Weinberg's discussion of cluster decomposition and localized particles in relativistic theories

As far as I know, in QFT it is very hard to make sense of a position operator. In fact, there is one thread on this matter here: "Position operator in QFT". Apparently the closest thing to the poisition operator is the Newton-Wigner position operator but if I understood it is not a fully satisfactory replacement. In the end of the day one simply works in the momentum representation.

Still there is one particular situation in which it appears necessary to talk about "localized particles" and that is when we talk about the cluster decomposition principle. Quoting Weinberg’s "The Quantum Theory of Fields":

We have formulated the cluster decomposition principle in coordinate space, as the condition that $S_{betaalpha}^C$ vanishes if any particles in the states $beta$ or $alpha$ are far from any others. It is convenient for us to reexpress this in momentum space. The coordinate space matrix elements are defined as Fourier transforms $$S^C_{{bf x}_1′{bf x}_2’dots {bf x}_1{bf x}_2dots}=int d^3mathbf{p}_1’d^3mathbf p_2’cdots d^3mathbf p_1d^3mathbf p_2cdots S^C_{mathbf p_1’mathbf p_2’dots mathbf p_1mathbf p_2dots} e^{imathbf p_1’cdot mathbf x_1′}e^{imathbf p_2’cdot mathbf x_2′}cdots e^{-imathbf p_1cdot mathbf x_1}e^{-imathbf p_2mathbf x_2}cdotstag{4.3.8}$$

Now this would make sense if there were a basis $|mathbf{x}rangle$ in the one-particle state space representing a particle localized at $mathbf{x}$. This is then tantamount to the existence of a position operator which, as I have already mentioned, seems to be very subtle.

In the absence of a position operator and a position basis, I can’t see why this is the right approach to talk about localized particles. Because of course if we want to say that "any particles in the states $beta$ or $alpha$ are far from any others" we must be able to say that such particles are localized and very distant to each other.

One last comment is that, if I understood correctly, the eigenstates of the Newton-Wigner position operator are not connect to the momentum eigenstates by the naive Fourier transform, meaning that instead of using the measure $d^3mathbf p$ one should employ the Lorentz invariant measure $d^3mathbf p/2 omega(mathbf p)$. This seems to add to the confusion, since $(4.3.8)$ seems to be employing the simple Fourier transform with measure $d^3mathbf p$ as in non-relativistic QM.

So how does one deal with particle localization in QFT in order to make precise the cluster decomposition principle? Why the simple Fourier transform is enough to deal with this, given that it appears no well-defined position operator can be defined in the relativistic setting?