Private Driver from Stuttgart (area) to Regensburg

Stuttgart and Regensburg are not one region - even not the same German state. Therefore it is unusual to use a driver / taxi for this route. Price would be extremely high.

I recommend to check train connections with DB journey planner:
https://int.bahn.de/en

This is a longer drive then it might seem. Definitely not a favorite of mine to drive. A6 Has gotten better though. Secondary roads dont help. Sorry can't offer an alternative. As suggested check the train routes.

We haven't used either of these in Germany, but we have used them in the UK, Italy, and Belgium:

https://www.blacklane.com/en/

https://daytrip.com/

You're looking at a 4 hour drive anytime during the normal workday. Might be faster on Sunday. I don't know any service other than the train outside of driving yourselves.

And it is 3 1/4 to 3 3/4 hour by train with a single connection in Nuremberg or Munich.