No, that's not sufficient. Since the observables are represented by self-adjoint linear operators on a Hilbert space which for some observables is infinite-dimensional, there is an infinite number of states for each particle, so Graham's numbers are insufficient.
As the wikipedia page mentions, physicists who work in the field and who believe there are hidden variables agree that experiments show these must be non-local.
You are confusing a model with reality. We have no way to observe the difference between a sufficiently large number and infinity.
For an overly pedantic counter example, each particle could simply simulate the rest of the universe to some finite precision.
PS: That's not to say you can't rule out specific theory's that use local variables. And we should use the simplest theory that works, however it's counter productive to suggest we can rule out all forms of local variables.
Nope. Entanglement has been observed for particles that have never coexisted in time [1]. This would mean that even your overly pedantic counter example requires that we give up free will: since we choose what property to measure, the first particle is unable to simulate the outcome of the second measurement unless it can simulate what we choose. (And yes, it's generally accepted that assuming superdeterminism, aka. no free will, allows a local hidden variable theory.)
In other words it still works. Note, it needs not be 100% accurate simulation. If it's good enough we can't tell.
Not that I think reality works this way, but the goal is to look for ways that a theory falls down, not look for evidence in support of a theory.
PS: FTL communication also works as a means to sidestep the need for a lot of quantum weardness. Sure, we don't like it but that does not mean it can't be happening.
As the wikipedia page mentions, physicists who work in the field and who believe there are hidden variables agree that experiments show these must be non-local.