You are a very lucky man, because you have found in RogueTrader a connoisseur of the Texas/Mexican border!
Yes, crossing the border is feasible. By car on a Friday afternoon is the busiest time, though. Plan on about an hour delay just to be safe. Usually, it takes about 20 minutes when its busy, and when its not busy there is no waiting whatsoever. But, allow an hour so you will be sure not to miss your flight from BRO, which I assume will be on Continental?
BRO does have long term parking, but I don't exactly how much it costs because I usually fly to HRL now. I think about $8-$10 a day.
The best thing for you to do would be to get dropped off at the border, cross the Rio Grande bridge on foot, and then take a cab from the American side to the airport. This would cost about $20, but you avoid the hassle of driving a car across and then paying to park it at the airport. Walking across the border is very fast, usually no waiting whatsoever.
While in the area you should visit South Padre Island: a beach vacation resort.