What specifically is the problem?
Using git for everything is not a bad idea because then you have consistent ids for each version and you can manage everything together.
You could build something that can manage multiple versions in SQL with an approval process (particularly if the order of the rows doesn't matter) but changes there won't be synchronized with git without more trouble.
Look for little things you can do to streamline the process you already have, for instance you might be able to hide the git manipulation behind some scripts if you haven't already.
Given that you want this to be part of PR/deploy cycle, I'd imagine most standalone options would not work. But you can significantly simplify your existing process.
1. Keep the primary copy on server somewhere -- in google sheets, or in Airtable, or even as Excel file on a shared drive. Make sure it has backups (or can be easily re-generated from csv in github repo)
2. Create an automation (github action, jenkins job, whatever) that takes current version of spreadsheet, exports to CSV, and makes a PR with new contents (and possibly submits for review)
3. In normal case, business employee changes primary copy, then starts the automation to create a PR. They can monitor created PR for any errors and/or merge progress.
4. There is a chance of accidental massive damage, like someone deleting the entire column by accident or restructuring data in incompatible way. That's when you step in and manually restore from backup/from csv file in github.
5. There is no locking, two business users might step on each other toes. But you say "only need to update occasionally" so this is hopefully fine for you...