Trying to assign the same degree of belief to infinitely many mutually exclusive options doesn't work.

Yeah, but the class of observers in the Doomsday argument is not infinite, usually one takes a small and a huge set, both finite. So in theory you could assign a uniform distribution.

For example, if you believe that universe has a limited amount of particles and a limited amount of time, that would put an (insanely generous) upper bound on the number of observers in this universe.

Exactly, and that's an assumption I'm always willing to make, to circumvent the problem of an infinite class of reference.

The problem though is not the cardinality of the set, it's rather the uniformity of the distribution, which I think is what is implied by the word 'randomness' in S(S|I)A, because I feel intuitively it shouldn't be so, due to the very definition of observer.