Does 'special relativity + newtonian gravity' predict gravitational bending of light?

It seems to me that special relativity (SR) already predicts that gravity will bend light rather than this effect being a proof of general relativity (GR). Photons have energy proportional to their frequency and according to $E = mc^2$ they also have a tiny, but non-zero relativistic mass equivalence. I have read the semantic argument that gravity deals with invariant or rest mass, but this should apply to a hypothetical photon at rest, not to real photons at velocity $c$.

I have considered the possibility that the SR effect is much smaller than the GR effect. GR explains the equivalence of inertial mass and gravitational mass as inherent rather than being a puzzling coincidence, but it is true in Newton's gravity, SR, and GR, so the quantitative difference between GR and SR does not seem right.