Which compose? Podman-compose [1] wasn't fully up to date with the latest compose-spec [2] the last time I checked it. However, the docker-compose v2 [3] (the one in Go, not Python [4]) is compatible with the Podman engine [5] and works like a charm for me.
I have also had no issues with networking, permissions or volumes while running as non-root user. Are you simply facing issues setting it up, or are you hitting some bugs or missing features?