For an almost drop-in daemonless, rootless docker replacement for this use case, see podman (https://podman.io/). You can `alias podman=docker` and it will just work.
You do need a bit of configuration, specifically you need to create `/etc/subuid` and `/etc/subgid` if they don't exist and add subordinate UIDs/GIDs which will be used to map the containers users and groups. E.g.
usermod --add-subuids 100000-165536 $USER
usermod --add-subgids 100000-165536 $USER