Good idea, but don't destroy the rental car plus on a Saturday there might be a regular race. A day in Basel might also be a good idea and if you really only want to drive: Drive down to Lake Constance, the A81 from Stuttgart to the south is the Autobahn with the least traffic in Germany, you'll often see people from abroad there that test drive Porsches and Mercedes sports cars, going from Stuttgart to the lake and back.