You could rent a car in Venice and turn it in Brunico, just down valley from Corvara. The only biggie I see located in Brunico is Avis, and their only rental location is Veice airport. So you could rent after you land, drive into Venice and park in the garage overnight while you relax in Venice. Just do not exceed the speed limit on the causeway by any amount as it is blanketed with speed trap cameras. Then drive to Corvara via Cortina and over the Paso Falzarego. Interesting stop there if you have any WW I interest, or want to take the cable car up for the view.
When you leave Corvara, drive down to Brunico/ Bruneck and turn the car in. Note they are closed noon Saturday until Monday morning. A short walk from the Avis location is the Bruneck Nord train station where you can get a train to Mittenwald, via Fortezza, Brennero, and Innsbruck.
PS from Pittsburgh?