There is really no need to stay in BZ City the night before your departure unless you want to. The flights to BZE star at 7AM so you can take an 8 or 9AM one and get there in plenty of time for your 11:30 departure with the 2 hr check in time required.
If you want to go over early and see a bit of the city, I recommend The Biltmore (closest to the airport), The Raddison, The Great House as potential places to stay.