The answer is that there are only a finite number of possible states that a Hubble volume can have, according to quantum theory
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Although classical physics allows an infinite number of possible states that a Hubble volume can be in, it's a profound and important fact that quantum physics allows only a finite number.
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Let's first ignore the important complication of past history and ask how many physically distinct states N there are in a volume V. In classical physics, N is infinite (indeed uncountably infinite) as you say, since even specifying the position of a single particle requires infinitely many decimals. In quantum mechanics, however, N is finite: if the temperature never exceeds T, we of course have N < ln S, where S is the entropy of the thermal state with temperature T (I'm taking Bolzmann's constant k=1). Interestingly, the number of states appears to be finite even when taking general relativity into account, which is closely related to the holographic principle: the entropy is maximized if all the matter in V is in a single black hole, in which case, as you know, the Bekenstein-Hawking formula says that N is of order the surface area measured in Planck units. So yes, I see your reasoning, and find it quite striking that quantum mechanics, uncertainty principle and all, contrary to what one might expect, gives fewer states than classical physics. In the limit V->oo, quantum mechanics therefore gives a countable rather than uncountable infinity of states
