Plausibly there is such a thing as a true physics, an ideal physics that we are striving towards, a physics that includes both particular statements as well as laws. That true physics is true, and hence consistent. It is also the sort of theory we can produce, so it has countably many statements (maybe finitely, but perhaps we could continually add to it). Finally, it is very likely to be a first-order theory, since it looks like all of science involves first-order theories.
Suppose there is a spacetime. Then the true physics posits it. Imagine we now have the true physics. By downward Löwenheim-Skolem, the true physics is consistent with spacetime being countable. So, by Ockham's Razor, wouldn't we have good reason to think that spacetime is countable, since that's more parsimonious than its being uncountable? And, now returning to the early 21st century of the actual world, doesn't the fact that if we had the true physics we would have good reason to think spacetime is countable, give us good reason to accept the counterintuitive conclusion that spacetime is actually countable?
I am not sure. For even if a countable spacetime is ontologically simpler, its description in our mathematical language is more complex than that of an uncountable spacetime manifold. Does that matter? Yes, but only if our mathematical language actually cuts mathematical reality at the joints or if we are created to get science right. (I suspect a naturalistic physicist could have a hard time resisting the argument for a countable spacetime. But this is all very, very speculative, and my mind fogs up when I think about the relativity of uncountability, intended models and the like.)