Would it? What does “stationary” mean when discussing relative velocities? The mirror being stationary and the person firing the photon moving at a constant velocity is literally an indistinguishable scenario from a stationary person firing the photon at a moving mirror.

If I am moving relative to a mirror when I fire the photon, then the mirror is moving relative to me, and will be in a different relative position by the time the “event” of my firing that photon reaches it.

Also, the photon isn’t moving infinitely fast in my (the firer’s) reference frame. It’s moving infinitely fast in it’s own reference frame.