Very interesting. I also enjoyed your previous paper I found on the arXiV (Right about Time). I had not previously made the connection between countability (one to one relation with the integers) and the continuum problem. Of course, you are right, its all about manifesting discreteness.
However, I would appreciate your opinion other aspects of countability. Consider, for example, the distinction between identity and indistinguishability. If we can identify particles as being discrete, but they are indistinguishable (for example in a Bose-Einstein condensate), then are they countable? More generally, do we need "identity" before the notion of countability is valid?
Also, what about recurrence? Imagine a pendulum (or other simple harmonic oscillator). If there is no physical mechanism to count beyond a modulo (in this case 2), then there could be an infinite number of recurrences (cycles of oscillation) before the event we observe, and after the event we observe. Can we refer to them as countable?
More importantly, if you are asleep, and someone wakes you, and you see the oscillator make a few cycles (you can even count them), and then you sleep for a while, but you don't know how long you have slept and someone wakes you again, can you still say that the number of cycles is countable -- if you have absolutely no external mechanism whatsoever to count the cycles that went by when you were asleep -- can you still say that the cycles are countable?
These may be easy questions for you to answer, nonetheless, they are interesting questions to me regarding "Is Spacetime Countable". For example, we could just as easily say "are cycles countable in the quantum domain"?
Kind regards, Paul