How does gauge invariance protect the SM gauge boson masses in SUSY from divergent radiative corrections?

What I say below are very general facts and probably this is not the final answer you were looking for but maybe it helps.

A gauge theory (forget about SUSY for the moment) gives rise to a massless spectrum of gauge bosons and massless matter content. If you want to give mass to your gauge bosons you need spontaneous symmetry breaking terms in your lagrangian (this means the absolute minimum of your potential is not unique). Furthermore, if you want matter particles to be massive you need to add Yukawa terms to your lagrangian. Assuming there is no spontaneous symmetry breaking one says "the maslessness of the gauge bosons is protected by gauge invariance" because you an explicit mass term would violate gauge invariance.

If SUSY is present but is not broken then your spectrum will be richer but again as long as your gauge invariance is not broken there is no reason to expect massive gauge bosons neither gauginos.

Now, what happens when SUSY is there but you break gauge invariance?

What happens if you break both SUSY and gauge invariance?

I am sorry but I do not know the answer to any of these...it seems to me that if only gauge symmetry is broken your scalar fields (and superpartner) will pick up vacuum expectation values in such a way that all particles and super-particles have the same mass. So you have masses but they have to match.

In the second case I guess you will have massive spectrum but the masses of particles and superpartners will not match.

Sorry I was not more helpful :(