What language classes beyond NP allow constant-round zero-knowledge proofs?
The sumcheck protocol can be made non-interactive, both in the random oracle model and under strong cryptographic assumptions. Good starting points to read about that are this paper and this paper. The high level intuition that even if Fiat-Shamir fails in general for interactive protocols with many rounds, it holds whenever the protocol satisfies some stronger notion of round by round soundness which the sumcheck protocol does satisfy.
This gives you a non-interactive argument for $\#\mathsf{P}$ in the ROM. I believe that it should be relatively easy to make it zero-knowledge. It might extend to all of PSPACE, though one would have to check the details - I do not think it has been written formally anywhere.
Edit: An important point, though, is that this result in the ROM will not be easy to extend to the standard model, at least not with standard cryptographic assumptions. This is because to get a constant-round argument for a language outside of NP, we would need a cryptographic assumption (for the soundness) which withstands even attacks from prover which are given the power of this larger complexity class, and this is typically not something given by standard cryptographic assumptions.