I am indeed denying the existence of photons interacting with virtual photons. I am also saying there is no tree level photon-photon interaction of on shell photons. Neither Compton scattering nor Bhabha nor pair production nor pair annihilation involves a photon-photon interaction. There is no photon-photon vertex in QED. There is no tree level Feynman diagram that you can look at and say “this is, at least philosophically, a photon changing its momentum”.
There is a 1 loop diagram that represents photon-photon scattering. But even that doesn’t have any photon-photon vertices, instead it is mediated by electron-positron pair.
Non-abelian gauge bosons (gluons) couple to themselves. So does gravity (gravitons). Abelian ones (photons) do not.
Photons don’t accelerate. They are emitted or absorbed. That’s their only interaction.