$\color{red}{\large \text{(For the mathematics & physics: see example at end!)}}$
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.
**(EDIT) PS:** I see that you now edited the question to address this fourth assumption, about the collapse. When you write "thus splitting reality into four branches", that may be a bit suggestive. The situation is a superposition, which is a normal concept in QM, we never call that a "splitting of reality" in other cases. As for your question about circumventing Bell, we now have, for the 4 branches after the measurements:
1) The QM state of the two maximally entangled particles clearly contained all necessary information to create the 4 complex amplitudes after the measurements, for every possible choice of the detector axes of A and B. (See below why this does not require infinite information!)
2) This information was in their combined entangled state long before any measurement was done and before the particles reached their (possible very distant) positions. So if the detector axes were changed at the last moment this did not lead to a special need for (superfast) communication, the necessary information was already there.
3) Although this QM state is nonlocal (describing joint properties of distant particles) it does not need interaction, or action, at a distance. So in the sense of your assumption 1), it is local.
4) Such a QM state does *not* contain enough information to select one of the 4 branches to be the "chosen one" after the measurements, only enough to give them the 4 complex amplitudes. And complex amplitudes can give quantum correlations that are higher than correlation between real positive probabilities which are restricted by Bell's theorem. So that is the answer to your question: describe things with amplitudes, not probabilities!
As for the precise information in the state: If Alice uses a basis $|0\rangle_A, |1\rangle_A$ and Bob $|0\rangle_B, |1\rangle_B$ then the Schmidt decomposition of a maximally entangled state (like the singlet state or one of the other Bell states, or certain linear combinations of them) can always be written as:
$$
\begin{align}
|\psi\rangle = \alpha\, |0\rangle_A|0\rangle_B + \beta\, |0\rangle_A|1\rangle_B
+ e^{i\phi}\beta^* |1\rangle_A|0\rangle_B -e^{i\phi}\alpha^* |1\rangle_A|1\rangle_B
\end{align}.
$$
Let's now assume that for whatever reason, Alice instead uses basis $|0\rangle_{A'}, |1\rangle_{A'}$ and Bob also changes his basis, we can still write
$$
\begin{align}
|\psi\rangle = \alpha'\, |0\rangle_{A'}|0\rangle_{B'} + \beta'\, |0\rangle_{A'}|1\rangle_{B'}
+ e^{i\phi'}\beta'^* |1\rangle_{A'}|0\rangle_{B'} -e^{i\phi'}\alpha'^* |1\rangle_{A'}|1\rangle_{B'}
\end{align}
$$
only it now has other coefficients based on altered values $\alpha', \beta',$ and $\phi'$. The new set of 4 amplitudes in the superposition can be obtained by just doing the appropriate (double) rotation in the space of states, so it requires no addition of infinite information for infinitely many choices of the detector axes that A and B might use.
$\color{red}{\large \bf PPS:}$
As suggested in [one of the other answers][4], let's assume the Bell state $|0\rangle_A|0\rangle_B+|1\rangle_A|1\rangle_B$ is the input entangled state and that
A and B then measure with 60 degrees axis difference, rotated in the $z-x$ plane. For spin-$\tfrac12$ states this gives the Schmidt decomposition:
$$
|\psi\rangle = \tfrac14\sqrt6 \,|0\rangle_A|0\rangle_B + \tfrac14\sqrt2 \, |0\rangle_A|1\rangle_B
+ \tfrac14\sqrt2 \, |1\rangle_A|0\rangle_B -\tfrac14\sqrt6 \,|1\rangle_A|1\rangle_B
$$
If A and B repeat the proces $N$ times we end up with
$$
\begin{align}
|\psi\rangle^{\otimes\,N}
&= \Big(\tfrac12\sqrt3 \ \frac{|0\rangle_A|0\rangle_B-|1\rangle_A|1\rangle_B}{\sqrt2}
+\tfrac12 \,\frac{|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B}{\sqrt2}\Big)^{\otimes\,N} \\
&= \big(\tfrac12\sqrt3 \ |C_+\rangle + \tfrac12 \,|C_-\rangle\big)^{\otimes\,N}
\end{align}
$$
where the normalized $|C_+\rangle$ and $|C_-\rangle$ denote the combined terms with matching and non-matching readings in
one iteration. After $N$ measurements, creating $2^N$ branches in the superposition, we can then evaluate this tensor product
and find those superposition terms where exactly $n$ of the $N$ results have matching outcomes (and $N-n$ have opposite outcomes).
There are $\tbinom{N}{n}$ of those terms, and each has amplitude $(\tfrac12\sqrt3)^n\ (\tfrac12)^{N-n}$,
and they are all orthogonal in the tensor product space. So if we add them up and call the combined sum term $|\Psi_n\rangle$, then it has amplitude:
$$
|| \Psi_n || = \binom{N}{n}^{1/2} (\tfrac12\sqrt3)^n\ (\tfrac12)^{N-n} \tag{a}
$$
The total superposition can then be written as:
$$
|\psi_{\,\text{Tot}}\rangle = |\psi\rangle^{\otimes\,N} = \sum_{n=0}^N \ |\Psi_n\rangle,
\tag{b}
$$
nicely splitting it up into terms where A and B have matching results in exactly $n$ of the $N$ iterations.
Now let's observe that the amplitudes $(a)$ are precisely the square roots of the binomial expansion terms in:
$$
\big(\tfrac34 + \tfrac14\big)^N = \sum_{n=0}^N \
\binom{N}{n}\ (\tfrac34)^n\ (\tfrac14)^{N-n} \tag{c}
$$
and we know that those terms get a very sharp peaking around $n=\tfrac34 N$, so their square roots will also have a peaking around that $n$ value, somewhat less pronounced by the square root, but in the large-$N$ limit still very strong.
So if (finally!) we want to see this as a many-worlds state (as was the intention in the other answer I referred to), then all terms in the tensor product state are branches, and we see that there is a very high peaking in amplitude for the subset of worlds where
Alice and Bob have their number of matching results, $n$, very close to $\tfrac34 N$. As is to be expected in QM (the Born rule rederived! QED).
[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
[4]: https://physics.stackexchange.com/questions/811135#811235