Well, the Sawgrass Mills mall is literally across the street from the BAC. It's a big outlet mall which has some restaurants, a movie theater, etc. There's a Cheesecake Factory in the outdoor section of the mall itself, and several other places right on that drive you take to go around it (Bahama Breeze, some fast food places, Moe's, and more). Avoid the sushi restaurant that's near there because the stuff I had was just not very good. I can't think of anything club-like right nearby, but it's not an empty area by any means.
What I do is go to the mall before the game and then to a restaurant after. You usually see people in jerseys at the mall, especially the away fans because I guess they come to see what's in the area too. If you can't sit though the game without food and are on a budget, grab something before you go in because the food at the BAC is kind of ridiculously priced. For fast food, I'd rec Pollo Tropical because it's something you guys don't have up in Baltimore. Or try Popeye's inside the mall for the Southern chicken, rice & beans, biscuits, etc. For a real restaurant, I really like Bahama Breeze but I haven't been to the Sunrise location.