[anonytrary]

> Light can reflect off of transparent crystals

If light is reflecting off of something, it's precisely because that thing isn't transparent[0].

> if the mirror is moving towards or away from the light, the reflection will be Doppler-shifted to a higher or lower frequency.

This does not refute OP's assertion. If the photon is absorbed and a "different"[1] one emitted, the emitted photon is still emitted from a reference frame that is moving with some velocity, therefore the emitted photon will still be Doppler-shifted... The photon is still emitted at the correct energy level. The photon appears to have higher energy in the stationary reference frame.

[0] Disregarding human-centric definitions that have to do with visible spectrum.

[1] Thinking of the photons as "different" doesn't actually mean much when it comes to quantum mechanics; they're indistinguishable bosons.