@karolherbst@chaos.social most docker-based ci systems (woodpecker, etc) won't allow you to start a whole docker compose out of the box. You would need to set up docker-in-docker or else do some hack like allowing the ci container access to the parent docker machine.
If you want a minimal solution to this, a scheduled task that pulls the repo and runs the build script if there is a new commit sounds good.
In terms of a proper ci system that has VM build agents, I'm not sure what to recommend. All the "modern" or "minimal" ones seem to use docker... So.... Jenkins ? Lol