Dear Rafael,
you present a bold argument for a radical sort of finitism. You argue your positions well, and present complex concepts in an approachable manner, without sacrificing too much accuracy.
The initial part of your essay reminded me of Baez' 'Rosetta Stone'-paper, where he develops the equivalence (at a categorical level) between computations and physical processes in great detail (if you haven't read it, I think you might find it enjoyable).
Another association triggered by your essay is with Jürgen Schmidhuber's 'Algorithmic Theories of Everything'. In particular, you write that there is no reason to really expect that nature should choose less complex realizations over more complex ones---but Schmidhuber argues that there is: assuming the probability distribution from which the history of the universe is sampled to be formally describable, he shows that it is dominated by those universe having a short formal description, i. e. a low Kolmogorov complexity.
Of course, neither of them shares your finitist commitment. I have to say that, while you argue your case well, this idea sits somewhat uneasily with me. For one, it breaks the quantum mechanical superposition principle---if amplitudes are only, at best, allowed to take on rational values (which one might say has no empirical consequences, the rationals lying dense in the reals after all, and possibly still 'dense enough' with sufficiently large numbers for numerator and denominator), then certain superpositions of states would be normalized to irrational values, i. e. infinite-precision quantities. Those would then have to be 'cast out' from the spectrum.
The problem also exists in a similar way within quantum field theory, which can be formally written as a continuous infinity of harmonic oscillators at each space-time point.
Of course, these might turn out to be mere approximations---but it's not immediately clear to me that one can modify the theories appropriately without breaking too much.
Still, I think it's an idea worth pursuing---and certainly, arguments (such as the one based on Bekenstein-Hawking entropy) that a theory of quantum gravity will only have a finite number of degrees of freedom within a given spacetime volume have been proposed in various ways.
So, best of luck---both to your program, and in this contest!
Cheers
Jochen