In your analysis you frequently mention "wave function collapse". But there is no reason to assume such a thing exists. Instead we can assume there is only subjective collapse. By insisting that it occurs you are limiting yourself to ["objective collapse theories"][1]. You are ruling out the many-worlds theory and probably some similar ones that go by other names.

When you describe assumption 2 (about the hidden variables) you introduce this new, hidden assumption: that wave functions can collapse!! So you should add this as assumption 4 (or perhaps better assumption 0?). You then of course do end up with about twice as many cases for selectively dropping some assumptions...

I also think the concept of "hidden" variables is not clear, since every theory has variables describing the state, and you can always call them hidden as long as you haven't measured anything. So what does a variable have to do to be called hidden, or not hidden?

Actually, for Bell's result, I do not think you really need all these assumptions since it is essentially just an upperbound on the correlation that is possible in joint probability distributions. Of course only if you sample them fairly, which looks like your assumtion 3, and of course there must be some information present, which you can always call "hidden" so we can drag in assumption 2, but assumption 1 certainly is not needed. Probability distributions are pure mathematical concepts, they do not need space and time to exist.

**Now to your question:** ways to get around Bell's theorem. I'll mention just one, using quantum mechanics without collapse!

So we drop assumption 0 (the collapse), we can keep assumption 1 (local QFT is our best theory!) we can forget assumption 2 since it assumes collapse to happen and we ruled that out. We can keep assumption 3, but reformulate it to: "the variables of the measured state are unrelated to the measurement device's measurement settings" (to stress that we do fair measurements but avoid mentioning the hidden variables of assumption 2 which we dropped).

Sticking to QM without collapse you will then get correlations that do violate the limit of Bell's theorem, but by at most a factor $\sqrt{2}$ because they still obey the [Tsirelson bound][2]. To get even larger correlations you could look at Popescu and Rohrlich's [PR box][3], but then we're going even further than what you ask, beyound quantum mechanics.


PS: I see that you now edited the question to address this fourth assumption, about the collapse. Good!



  [1]: https://en.wikipedia.org/wiki/Objective-collapse_theory
  [2]: https://en.wikipedia.org/wiki/Tsirelson%27s_bound
  [3]: https://en.wikipedia.org/wiki/Quantum_nonlocality#The_physics_of_supra-quantum_correlations