If that's the purpose, couldn't you store the hash and throw away the compressed image?

(As others said, compression is deterministic for the same algorithm, parameters and input data)

Recompressing should be guaranteed deterministic. It’s the packing/unpacking of tar archives to/from directories on disk that leads to the non-determinism (such as timestamps and ownership metadata). If the tar is left intact, both zstd and gzip should produce byte for byte identical outputs given the same compression parameters.