I guess first you'd have to prove that there are physical processes are actually continuous, and that this continuum is genuinely the continuum of the reals.
It is already known that a computer with access to real numbers with infinite precision can perform hypercomputation, I'm not sure how much this is relevant, but:
In standard physics, the space and time are continuous, and nothing else. All the physical quantities that are derived from length or angles or time are continuous (this includes quantities like mass and energy, which are derived from time). Besides those, there are no other physical quantities that are continuous.
The mathematical models that assume the continuity of space and time match extremely well any experiments and for now there exists no evidence whatsoever that space and time are not continuous.
Besides the standard physics, there have been a few attempts to create mathematical models of the physical processes where the space and time are discrete. Nevertheless, these mathematical models did not succeed to demonstrate any advantage over the standard models with continuous space and time.
I agree with you, but is there any peer-reviewed publication that can be cited? The idea makes sense to me, firstly the Reals \ Inaccessible Reals = Computable Reals, secondly you can't ever input an inaccessible real to an experiment nor retrieve one out of an experiment -- but then I'm not completely certain in making the conclusion that no experiment can be devised which shows that inaccessible reals exist in physical space.
I am concerned about this in the field of complexity analysis of quantum computers too, I think that the use of reals in physics is leading to mathematically correct but non-physical results about complexity theory of quantum computers. Having a paper to point at and say "look, stop assuming your Bloch spheres are backed by uncountable sets, it's leaking non-computable assumptions into your analysis of computation" would be helpful.
I took an interest in this area once, and this is not true. "Computable analysis / computable topology" says quite the opposite.
https://en.m.wikipedia.org/wiki/Blum%E2%80%93Shub%E2%80%93Sm...