Bell's theorem is strong: No local hidden variables can explain quantum entanglement.
Your proposed fifth dimension is exactly such a variable – there is no set of values it can take on that explains the quantum phenomena we can easily and reliably observe.
The fact that you think of it as a distance itself isn't important to Bell's theorem. Such a theory is cannot even obey relativity in our existing 3 dimensions (obeying the speed of light in your extra dimension would be even stricter!)
Sure. There are assumptions you need for Bell's Theorem to apply. Statistical independence, single-outcome measurements, locality, determinism... All of these feel right to our scientific brains, but something's got to give.
So yes, you can throw out Bell's theorem if you're willing to throw out statistical independence with super-determinism. It's a tough leap to make, though. Super-determinism implies that there aren't really any laws of quantum mechanics. The fact that billions of quantum interactions have been observed in labs to obey consistent probabilities over time is pure coincidence. The fact that two entangled electrons always have opposite spin is pure coincidence. Nothing about the laws of the universe say you couldn't observe the other thing, it's just that the initial conditions of the universe prevented us from making measurements any time we would have measured otherwise. It's a tough pill to swallow.
My comment above is just saying you can't throw out Bell's theorem by saying "maybe there's a fifth dimension". I don't think GP meant a fifth dimension plus superdeterminism.
Your proposed fifth dimension is exactly such a variable – there is no set of values it can take on that explains the quantum phenomena we can easily and reliably observe.
The fact that you think of it as a distance itself isn't important to Bell's theorem. Such a theory is cannot even obey relativity in our existing 3 dimensions (obeying the speed of light in your extra dimension would be even stricter!)