We build DoltDB, which is a version-controlled SQL database. Recently we've been working with customers doing exactly this, giving an AI agent access to their database. You give the agent its own branch / clone of the prod DB to work on, then merge their changes back to main after review if everything looks good. This requires running Dolt / Doltgres as your database server instead of MySQL / Postgres, of course. But it's free and open source, give it a shot.

https://github.com/dolthub/dolt