Brussels and Garmisch are in direct opposite directions, so it would be an either/or decision, not both. Personally I'd pick Garmisch and the Alps. Garmisch is about a 3-3.5 hour drive from Stuttgart (Brussels is about a 6 hour drive).
Garmisch is a typical scenic Alpine town (and a little touristy, but worth it), and a popular ski destination. The Zugspitze is right next to Garmisch, and is the tallest mountain in the German Alps.
A meal at the Gasthof Fraundorfer in Partenkirchen is worth it - typically includes music and German dancers.
Since you mentioned Cantillon, you might enjoy the Andechs monastery/brewery. It's between Garmisch and Munich, if you elected to drive that way (or on the way back). Andechs Doppelbock Dunkel is one of my all time favorites.
As far as Stuttgart, both the Mercedes and Porsche museums are worth visits, particularly if you're a car guy. My son was born about 5 minutes away from the Porsche factory when we lived there.
Due to Stuttgart's industry, the city was extensively bombed during WWII, so it doesn't have as much of the old architecture that other cities are known for. But a walk down the Konigstrasse is worthwhile, as is the Schlossplatz in the town center.