I know that there is a mistake in reasoning but I cannot pinpoint it.
1) The standard response in all such situations is:
Draw the spacetime diagram.
2) Probably the conceptual error is to think that bouncing off a
moving mirror is the same as bouncing off a stationary mirror.
It's not the same. The energy is different and the direction is
different. The moving mirror does work on the light.
Make the analogy to playing pesäpallo. A ball hitting a moving
bat is not at all the same as hitting a moving bat.
If that's not enough of a hint, please re-ask the question.