Provided the connection is on Delta, I would pay the extra $200 and fly to Birmingham for several reasons.
1. Saving several hours (6 or so RT) of driving.
2. Easier baggage collection on arrival due to airport size.
3. Quicker check-in and security on departure due to airport size.
4. Miles due to Avis promotion verses miles from extra flown leg is probably a wash, but not the most important aspect of the decision.