Can one body push another body away by sending gravitational waves towards it?

Let us first discuss why do usual bodies absorb electromagnetic waves so well. A normal body is made of protons and electrons, which have specific charges of opposite sign and vastly different magnitude. As a result, every atom will be strained by the electromagnetic wave even if its size is completely negligible with respect to the wavelength because they feel opposing forces at any point. Zooming out, atoms and/or molecules will have a magnetic dipole moment, which can also accelerate it as a whole. If the body is made out of metal, free electrons react to the wave immediately, bump into the protons in the metal lattice, and cause heating. All these phenomena come about because the body is made up of elements that react differently to the passing electromagnetic wave, and this is also the reason why usual matter is such a good absorber of EM radiation.

Now let us take a look at gravitational waves. Here the situation is very different since every piece of matter in the Universe has the same "specific charge" with respect to gravity. (This is called the equivalence principle and we know it from our everyday as "gravity accelerates everything equally".) When the gravitational wave then passes through a structure much smaller than its wavelength, the entire structure is moved in sync. It is like a small boat on a tsunami wave far from shore: nothing really happens.

So if any absorption comes about, it will be from the fact that the body is slightly deformed on scales comparable to the wavelength. This causes internal friction and ultimately also some sort of GW back-reaction from the body. We can approximate the back-reaction by assuming that the energy dissipated by the internal friction $Delta E$ must be compensated by a change to the momentum of the body $Delta p = Delta E/c$ as you state in your question. So yes, some momentum transfer can occur for bodies of comparable or larger size than the wavelength and large internal friction.

However, even a pretty powerful wave will cause modest strains far from the source. It can be shown that a GW source of mass $M$ will cause a strain at most $lesssim GM/(c^2 R)$ where $R$ is the distance from the source (the upper bound is satisfied only for equal-mass binaries at merger). For example, if you wanted to cause a strain of $10^{-2}$ at a distance of $100$ meters, you would need a generating mass of the order of magnitude of $sim 100$ Earths (in a black hole binary near to merger). It is good to note, though, that no object of size smaller than $GM/c^2$ would ever be strongly affected by the wave since that is the lower bound for the wavelength.

About the "GW pressure" overpowering the gravitational attraction. It is good to again remember that gravity has only one sign of "charge". In electromagnetism, we can put a positive and negative charge together, make them oscillate, and the resulting oscillating dipole has no net attraction or repulsion for any single particle. This cannot be done in gravity, every source of radiation in gravity comes with an attraction that satisfies certain inequalities with respect to the radiation field. For instance, if you want to collimate your wave, you cannot do it without putting masses next to each other, there is no such thing as a "GW reflector". At this point I offer a conjecture anyone is free to explore: I believe that if you go into the question of how much momentum can be absorbed by the internal friction and put fundamental physicality bounds on the friction coefficients, the "GW radiation pressure" will never be stronger than the attraction apart from a special limiting configuration at the moment of the merger.

Short answer: Yes, it is possible to produce enough force from gravitational radiation to overcome gravitational attraction to the source. However, there seems to be some fundamental limitations that makes it impossible to transfer enough linear momentum via gravitational radiation to put the body initially at rest on an escape trajectory, and so, this “overcoming of gravitational attraction” is only transient.

My idea is based on the fact that large black holes (and other bodies for which the size is comparable to Schwarzschild radius), make efficient absorbers/deflectors of (shortwave) gravitational radiation with arbitrary small cross section to mass ratios. But despite such low surface density to move it one needs a large source of gravitational radiation heavy enough that it would not fall itself into the black hole it is irradiating. To achieve this goal we can arrange a very large number of relatively small independent emitters of GW in a nonrelativistic “swarm”. If such swarm is sufficiently sparse then the metric would remain almost flat throughout and linearized analysis is applicable. Therefore the peak luminosity of gravitational wave $mathcal{L}_text{GW}$ that could be achieved from such a system as well as its total mass would scale almost linearly with the number of individual components. And since both the Newtonian gravitational force and the gravitational radiation fluxes obey inverse square distance laws, one could make the swarm arbitrarily large in order to lift the supermassive black hole that would have small enough effective “surface density”.

Detailed analysis: As a starting point we assume that we can make an isolated and self contained emitter of gravitational waves with approximately fixed mass $m_0$ and gravitational wave luminosity $mathcal{L}_0$ with characteristic wavelength of gravitational radiation $lambda_0$. This could be as powerful as a pair of orbiting black holes during final stages before the merger or as weak as a $20,text{m}$ and $490,text{t}$ steel beam rotating around its center (Excercise 36.1 in MTW).

By replicating this GW source $N$ times and placing the copies in a swarm on nonrelativistic orbits around their common center of mass and with random orientations, we could make an arbitrarily large and nearly isotropic source of gravitational radiation with the mass $Mapprox N m_0$ and gravitational wave power $mathcal{L}_text{GW}approx N mathcal{L}_0$. We used approximate equalities because the total mass of the swarm is going to be modified by the kinetic and gravitational binding energy of orbiting emitters, while the power and wavelength of gravitational radiation would be modified by the Doppler effect and the gravitational redshift. But by increasing orbits' sizes we could always make the discrepancy arbitrarily small and overall metric close to Minkowski spacetime.

A body that we want to push away would be characterized by its mass $mu$ and an effective cross section $S$ describing gravitational wave pressure on that body. Newtonian gravitational attraction towards the swarm would be: $$ F_text{N}=frac{G M mu}{R^2} approx frac{G N m_0 mu}{R^2} ,$$ while the force from the gravitational radiation $$ F_text{GW}=frac{mathcal{L}_text{GW} ,S}{4pi R^2 c} approxfrac{N, mathcal{L}_0 ,S}{4pi R^2 c}.$$ Both of those forces obey the inverse square law, so we could make the distance $R$ arbitrarily large to minimize such effects as tidal disturbances of the swarm by the body and vice versa.

If the gravitational radiation pressure overcomes gravitational attraction, $F_text{GW}>F_text{N}$ the body is being pushed away. This occurs when $$ frac{S}{mu} > frac{4pi c G m_0 }{mathcal{L}_0}. tag{1}$$

Note, that the condition on the ratio $S/mu$ does not depend on the number $N$ of small emitters or on the distance between the body and the swarm of emitters. However, for the body to be moving away from the swarm, not only its acceleration must be directed away, but also gravitational acceleration exerted by the body on the swarm must be smaller than acceleration of the body itself. For this we need to choose the number $N$ of emitters large enough, so that the mass of the swarm exceeds the mass of the body: $$Ngg mu / m_0. tag{2}$$ (If the acceleration of the body and the swarm coincides we would have a bizarre case of gravitational wave propulsion: the body is pushed away by radiation from the swarm and at the same time pushes the swarm behind it).

For our body we could take a large Schwarzschild black hole. We could estimate the effective cross section $S$ with its capture cross section (see e.g. this question) $$S>sigma_c=frac{27}4 pi r_s^2,$$ where $r_s=2 Gmu /c^2$ is its Schwarzschild radius. The actual cross section for the transfer of linear momentum from gravitational waves would be larger since linear momentum gets transferred to the black hole even when gravitational wave is deflected without absorption. The conditions $(1)$ and $(2)$ would then read: $$ mu>frac{4 c^5 m_0 }{27 G mathcal{L}_0}= frac{4 mathcal{L}_text{P} }{27 mathcal{L}_0} m_0,qquad Ngg frac{4 c^5 }{27 G mathcal{L}_0} = frac{4 mathcal{L}_text{P} }{27 mathcal{L}_0}, tag{3} $$ where $mathcal{L}_text{P}=c^5/G$ is the “Planck luminosity”. So it is always possible to make the black hole absorber large enough that gravitational radiation it absorbs would produce enough force to overcome gravitational attraction from GW sources, and to choose $N$ large enough, so that the swarm would stay approximately in place while the black hole is pushed away.

Limitations: The appearance of Planck luminosity (we use the term though there is no Planck constant in it) is a hint that unless gravitational radiation is produced very efficiently, the number of emitters in the swarm would be very large. A steel beam from MTW exercise would have to be replicated $Ngtrsim 10^{81}$ times to push a black hole large enough to have the necessary surface to mass ratio. Such a swarm (and such a black hole) would exceed the size of observable universe by many orders of magnitude (and so would be unachievable even in principle because of the cosmological constant/dark energy). On the other hand, a pair of merging black holes could achieve $mathcal{L}_0sim 10^{-2} mathcal{L}_text{P}$ and so, if we use black hole mergers as a source of gravitational waves, the mass of black hole absorber would have to be just several hundreds times larger than the individual pairs, and the swarm could consist of about a thousand pairs to produce enough gravitational waves to push the absorber away and not to be pulled along with it.

Above analysis was only concerned with peak gravitational wave luminosity. But any emitter would have a characteristic period of activity of the emitter $tau$, during which it could produce gravitational wave luminosity of approximately $mathcal{L}_0$. Obviously, $tau mathcal{L}_0 < m_0 c^2 $ and using ($3$) we can obtain that $$frac{2 M G}{c^2} >tau c .$$ And since we assumed that the swarm is much larger than its Schwarzschild radius, we see that the period of activity of the emitter would be much smaller than the swarm light-crossing time. So while it is possible to produce enough gravitational radiation to overcome the gravitational attraction, we must do so by carefully timing the periods of activity of emitters so that the gravitational waves from all of the emitters would approach the absorber during the same time interval.

It also seems that the total momentum imparted by the gravitational radiation would never be enough to place absorber on the escape trajectory (assuming that it was either initially at rest or on a Newtonian circular orbit). This seems to be the fundamental limitation: total power transferred to the absorber must exceed the Planck luminosity, which seems to be impossible to achieve without formation of the additional horizons. For a discussion of Planck luminosity as a limit to a power within a gravitational process see the paper by Cardoso et al. which in section IV contains a discussion of a system similar to our swarm, also see this question on PSE. So we could formulate a conjecture: using only omnidirectional emitters of GW, it is impossible to transmit enough momentum to a black hole, or other absorber with comparable characteristics to put in on escape trajectory from the initial state of rest.

At large enough separation this pressure, small though it is, will nevertheless exceed the gravitational attraction if it falls off slower that $1/r^2$, which it will do if it is sufficiently directional.

In the above discussion we considered only omnidirectional sources of gravitaional radiation. But if the radiation has nontrivial directional pattern, can we produce a beam of gravitational radiation that would transfer the momentum to sufficiently far away absorber without forming a black hole first? If we simply tuned orientations of our emitters so that their radiation patterns maxima would be oriented towards the absorber that would only alter the coeffiecients in the equations without changing any of the limitations. In particular, the GW flux fall off would still be $1/r^2$. Also, if the pattern is asymmetrical, the emitters would produce overall thrust: rather than pushing the body away, the swarm of emitters would be moving away from it under own power. And in our setup, this last effect would exceed any push to the body.

We could hope that there exist an efficient mechanism of gravitational waves that scales faster than linearly with the overall mass. We could envisage that this “gravitational wave laser” could easily overcome the limitations outlined above. And indeed, linearized gravity calculations seems to suggest that under certain conditions this is possible, but again, can this scale up to a fully nonlinear system without problems (such as formation of horizon around would-be “laser”). Another avenue of improvement could be a more efficient absorber/scatterer of gravitational waves than a black hole. This could be combined with a more efficient emitter: resonant scattering of gravitational waves could transfer enough linear momentum to a lighter body than a black hole, but again this is still a conjecture.