congratulations on an eminently readable and engaging essay on a difficult topic! I will need some time to fully digest your arguments, but I wanted to leave a few preliminary comments---also because our two approaches have some overlap, in particular as regards undecidability and Bell/EPR.
I'll state my biases upfront: I'm skeptical of any sort of 'completion' of quantum mechanics by hidden (or not-so-hidden) variables, that is, viewing quantum mechanics as a statistical theory of some deeper level of reality, and I'm in particular skeptical of superdeterminism.
That said, I'm always happy to see somebody giving an 'alternative' approach a strong outing---and that you certainly do. I do see now that I had dismissed these topics perhaps too quickly, so I've got to thank you for that. But on to some more detailed points.
You note the similarity between the Liouville equation and von Neumann's equation; you probably know this, but that similarity can be made much more explicit by considering phase-space quantization. There, the Moyal equation emerges as a deformation of the Liouville equation (with deformation parameter hbar), and contains the same empirical content as von Neumann's (they are linked by the Wigner-Weyl transformation).
I'm of two minds whether this supports your contention, or not. On the one hand, you can explicitly link the quantum evolution to that of a stochastic system; on the other, the deformation by hbar essentially means that there's no finer grain to the phase space, you can't localize the state of the system any further.
I'd be interested in how your approach works with things like the Pusey-Barrett-Rudolph theorem, that's generally thought to exclude viewing quantum mechanics as a stochastic theory of some more fundamental variables---although, as usual, and as you seem to be adept at exploiting, there are various assumptions and caveats with all 'no-go' theorems. I think there's an assumption that successive preparations of a system can be made independently; I'm not sure, but maybe that fails, taking one out of the invariant set.
I'll also have to have a more thorough look at how your model system is supposed to yield Bell inequality violation. Of course, having a Hilbert space formulation is not sufficient---you can formulate classical mechanics in Hilbert space, too (the Koopmann-von Neumann formalism).
I've got to go now, I'll add more later when I have some time. In the meantime, congratulations on a very strong and interesting entry into this contest!