I've been through birmingham many times on business. The Sheraton is your best bet for proximity(short uber/cab ride away), as is any other downtown hotel. Personally I'm thinking of staying a little south of downtown at the Aloft, its a W hotel property and its pretty badass. Its listed as one of the affiliated hotels on the bowl page. It'll run you around $120-130 a night but its a posh W like experience. Sheraton is a stock convention center type with normal rooms for probs the same price. Aloft is about a 10-15min drive south, closer to five points.
just my $.02
GO BULLS!!!