I'm an amateur as far as SR/GR is concerned, but I do know that the correct interpretation is that light/photons simply follow geodesics in spacetime.
I fail to see any connection. Are you perhaps thining of Doppler effect and cosmological and gravitational redshift? That's a different ball game as far as I am aware.
Gravitational radiation are simply ripples in spacetime caused by catastrophic collisions and/or asymmetric S/nova explosions.
SR was brought to our attention in 1905 and since then hundreds of scientists have looked and studied it from all angles. I'm pretty sure if there were any inherent problems, we would know about them.