I've created a simple project for that, except that by default they open PRs on Forgejo/Github/Gitlab. It's open source and the workflows are config files, so you can configure it to allow you to pull from them instead of creating a PR. Source: https://github.com/smithy-ai/smithy-ai

One way I use it that is similar to what I think you want is to run a local Forgejo instance (just like OP) and have the Dockerised Claude instances work in that, then pull from there to my local git repo, review, test etc and push from my local repo to a remote like Github.