A couple things about that area.
If you have time, lunch in Bregenz is a nice option, specifically, lake front dining. https://www.wirtshausamsee.at/ Decent food, but a spectacular view. You can see three countries sitting there, Austria, Germany and Switzerland.
A short waterfront stroll away is Festspielhaus Bregenz.
It is a waterfront theatre with the stage built into the water. And it is always a very clever innovated stage changed yearly.

As far as driving one side or the other.... I have not driven or visited the north side of the lake. But we did drive along the Swiss Alps. The views are amazing. And a bonus, you could go a few miles out of your way and hit Lichtenstein. Not impressive, but it adds a country.
I will leave it to other more experienced travelers in that area to give their input on a northern route versus a southern route.
Oh, and the swiss motorway vignettes are easily obtainable at rest stops at the border.