The explanation I've read on why a free electron can't emit a photon goes like this:
Let there be a free electron of mass $m$ moving with constant velocity $v$. We may enter a new reference frame where the electron isn't moving. If the electron suddenly emits a photon of energy $E = hν$ and momentum $p = hν/c$, then by conservation of momentum the electron will recoil with momentum $hν/c$ as well, but in the opposite direction. This means the electron gains kinetic energy $K$. Let us write the conservation of energy in this frame. The energy before emission is $mc^2$ and the energy after emssion is $mc^2 + K + hν$. Therefore
$$mc^2 = mc^2 + K + hν$$
And this implies $K = -hν$. Since K and h are both positive, the frequency ν has to be either zero or negative. This implies:
$$ν = 0$$
And therefore no photon is emitted is this frame. Therefore, no photon is emitted in any frame.
Ok, this seems like a pretty good explanation. The step I'm having trouble with is the conservation of energy. What exactly stops the electron from being an additional source of energy? I mean, couldn't the electron lower its mass by $Δm$ (somehow) thus giving an extra energy $Δm.c^2$ for the generation of the photon? This would mean:
$$mc^2 = (m-Δm)c^2 + K + hν$$
$$Δm.c^2 = K + hν$$
And nothing appears to be violated. I have tried to refute this argument by using the kinetic energy of the electron as a function of its momentum. Sadly, the math didn't bring me to any contradictions. I believe something more fundamental about the electron's mass is going on here. What exactly is the mistake in this idea?