Set your pod to pull image always and have entrypoint shell script that clones the repo, kill the pod so on restart you could get your code deployed.

You could run init container with Kaniko that pushes image to repo and then main container that pulls that back but for that you need to do kubectl rollout restart deploy <name>

If you are looking for pure CI/CD gitlab has awesome support or you could do Tekton or Argo. They can run on the same cluster.