Baden-Baden
then heading South through the Black Forest (use all back roads)
to
Singen (Auto-Salon)
Konstanz, take car ferry to
Meersburg, have lunch there.
drive along Bodensee (Lake of Constance) to
Bregenz (Austria) from there go to
Appenzell (most rustic and authentic Kanton/Caounty of Switzerland) and have some cheese fondue or Racquelette (Cheese molten at a wood fire).
Then go to Thusis to enter Via Mala (translation "Bad Road" built by the Romans throught he Alps). You can do some sightseeing there.
then take the
San Bernardino pass to go to the
Italian Lake country of Lago Maggiore, Lago di Como etc, your pick, can't go wrong with either, best food you will ever eat.
From there head to
Chamonix and the Mont Blanc (you have to drive though the tunnel, amazing, hand cut through the rock, was for a very long time the longest tunnel in the world).
Start your loop back through Lausanne/ Lake of Geneva to
Bern and then Basel to enter back into the South of Germany.
I would take the Autobahn from Stuttgart to Baden-Baden, so you can check that off your list and then stay on county roads, which there should be one almost everywhere parallel to the highway. then when you come back get on the Autobahn in Singen and abck to Stuttgart, should all be unrestricted and fairly low traffic to blow out the end of your tour. Just distance wise you could probably do this whole tour in 3-4 days without sightseeing and stops.