What stops a free electron from 'using' part of its own mass to generate radiation?

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?

The mass of the electron is fixed. If you want to use some of it, then the electron must decay into another, lighter particle.

There are not very many particles that are lighter than the electron. Basically, there are the neutrinos (which are quite light, but have nonzero mass) and there are the massless gauge bosons (photons and gluons). An electron can convert to a neutrino by the emission of a W$$^-$$ boson, which, at a mass of roughly 80 GeV (157,000 times the mass of the electron!), cannot be emitted spontaneously except at very, very high kinetic energies, and certainly not while the electron is stationary. If the electron interacts with a passing neutrino, it can exchange a virtual $$W^-$$ boson and convert to a neutrino, but 1) this requires something else other than the electron to be present, which I believe is out of the scope of the situation you're asking about, and 2) has an incredibly low probability of actually happening.

An electron carries no color charge, so it cannot interact with, and hence cannot decay to, a gluon. Decaying to a photon would violate the conservation of charge. Hence, the electron cannot spontaneously change its mass, because the electron is stable. This is, to say the least, good news for all of the atoms in the universe.

The rest mass of an elementary particle is a fundamental property of the particle and cannot be changed. So an electron cannot change its rest mass without stopping being an electron i.e. decaying into some other particle. Since the rest mass is the mass in the electron's rest frame, and in your example we are working in the rest frame, that means the electron cannot reduce its mass to produce a photon.

An electron does not have the internal degrees of freedom that this would require. An atom in an excited state does.

As far as I am aware, electrons always have the same mass, and cannot reduce it - unless they could transform to other, lighter particles. Of course, as long as we discuss a rest mass, ie. a mass of a stationary electrone. The situation is different if an electron is moving, especially with speeds close to the speed of light. In such cases the mass is bigger by a factor depending on the velocity - and theoretically could be reduced by a mere slowing down. But the changes would be different depending on a frame of reference - that's probably the reason it's not observed.

Physical theories must conform to observation. And observations indicate that all electrons have the same (rest) mass.

To be more precise, we see (eg, in Millikan's classic oil drop experiment) that charge is quantized, and that all electrons appear to have identical charge, and identical charge to mass ratio. These things wouldn't be true if electrons could lose mass.

In statistical equations, electrons (like all fundamental particles in the Standard Model) behave like indistiguishable particles. This is a very important factor in our quantum theories. The world would be a very different place if this uniformity weren't present.