Is it theoretically possible for a proton to travel fast enough to destroy the sun by virtue of its relativistic mass?

TL;DR: Probably not.

The Sun has a lot of gravitational binding energy, roughly $3.8×10^{41}$ joules. Your proton needs to have kinetic energy in that neighbourhood, and it needs to deposit it deep inside the Sun to do much damage. From that Wikipedia article:

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy in order for hydrostatic equilibrium to be maintained.

When a star forms, gravitational potential energy is converted to thermal energy. Half of that energy stays inside the star as its internal thermal energy, the other half is radiated away. This is known as the Kelvin-Helmholtz mechanism. Lord Kelvin thought that it was the process that powers the Sun. However, that mechanism would only produce enough energy for the Sun to shine with its present luminosity for around 8.9 million years. Of course, the Sun is actually powered by nuclear fusion. Still, 8.9 million years worth of total solar output is nothing to sneeze at. And that's roughly how much energy your proton accelerator needs to supply to your killer proton. I suspect you'll face extreme difficulties in getting that much energy, and in pumping it into a single proton. ;)

Your plan is to shoot the KP (killer proton) at the Sun from a distance of 1 au, and for it to pick up additional momentum due to the Sun's gravity. There are a couple of problems with that. Firstly, interplanetary space isn't a perfect vacuum. It contains numerous protons and other particles from the solar wind. True, the density of those particles is small, but when any of those particles collide with the KP they will take some of its momentum & KE, essentially acting as a form of friction. And the Sun's magnetic field will also deflect the KP's path and act as another form of friction.

Secondly, the KP won't actually pick up much momentum falling towards the Sun. At low speeds, a body falling from 1 au to the Sun gains about 40 km/s, but that's insignificant to your ultra-relativistic KP, which has kinetic energy many orders of magnitude greater than its rest mass. At high speeds, you can't just add speeds together, you need to use the relativistic formula for composition of velocities: $$w = frac{u+v}{1+uv/c^2}$$