I have found this question here Can a free particle absorb/emit photons?, along with other resources that show a free particle cannot emit a photon (in a vacuum).
Now, I am 90% sure it does, but does this result hold generally, ie not just photons/ photons but in a medium?
If you go in the rest frame no matter what it emits energy is not conserved. The reason I am questioning this is that an exam question said 'if it can emit it find the angle relative to the motion of the electron'. Seems odd that they would give away marks for just repeating that it can't.