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There are only so many distinct states of experience, so yes, causal models are countable. The set of all causal models is a set of functions that map K n-valued past experiential states into L n-valued future experiential states.
This is a monstrously huge number of functions in the set, but still countable, so long as K and L are at most countably infinite.
Note that this assumes that states of experience with zero discernible difference between them are the same thing - eg, if you come up with the same predictions using the first million digits of sqrt(2) and the irrational number sqrt(2), then they're the same model.
But the set of causal models is not the set of experience mappings. The model where things disappear after they cross the cosmological horizon is a different model than standard physics, even though they predict the same experiences. We can differentiate between them because Occam's Razor favors one over the other, and our experiences give us ample cause to trust Occam's Razor.
At first glance, it seems this gives us enough to diagonalize models--1 meter outside the horizon is different from model one, two meters is different from model two...
There might be a way to constrain this based on the models we can assign different probabilities to, given our knowledge and experience, which might get it down to countable numbers, but how to do it is not obvious to me.