Probably the easiest if you just set up a git repo and configure a post commit hook to do whatever needs to be done, including moving the state from the repo to the web folder.
We do that with our Plesk servers all the time.
https://www.namecheap.com/support/knowledgebase/article.aspx/10113/2210/how-to-use-git-version-control-cpanel-plugin/ looks like you can configure deployment scripts with that.