Post-quantum SNARGs with near-constant verification?
There exists as of today two candidate SNARGs satisfying all your criteria. The first one is a recent proposal by Boneh, Ishai, Sahai, and Wu, which was presented last year at Eurocrypt. They achieve this construction by building upon the notion of linear-only encryption, which had been developed in previous works of some of the authors.

The second one, by Gennaro, Minelli, Nitulescu, and Orrù, is even more recent, it was presented at CCS this year. It generalizes square span programs, which were previously used to build SNARGs under pairing-based assumptions, to the lattice setting. It also comes with a proof-of-concept (unoptimized) implementation.

Both approach suffer from a common downside: they are designated-verifier, which means that a secret verification key is required to check the proof. If you crucially want public verifiability, even though it's theoretically feasible in a quantum-safe setting (as you mentioned, applying the random-oracle-based Fiat-Shamir transform to Kilian's succinct PCP-based proof system does the trick, and leads to logarithmic proof size and verification time), we do unfortunately not yet know of any concretely efficient construction that would meet all of your criteria.