The really puzzling part of this thought experiment comes when a “photon” interacts with an electron. If we could travel along with the photon then we would “perceive” the electron as approaching at the speed of light. But since the electron has mass, it can’t travel at the speed of light, c. Thus “we” can’t be traveling at the speed of light. This seems to produce a contradiction.
My suspicion is that the solution to the paradox involves the differences between the free field solutions of the photon which are infinite in extent (present everywhere simultaneously with a “pure” energy spectrum) and “real” photons (those produced in a lab). Real photons will be a linear combination of free fields, have a meaningful position in space and time, and will travel at a group velocity with c as an upper bound.
But it’s been a quarter of a century since I was last in a field theory course. I’ve forgotten far more than I remember. Hell, I may have forgotten more than I ever knew!