If you have the time, it may be worth the trouble to get over to BGI if you want to save a bit. A fares ex-BGI make NAS look like extortion... and you can do the same routing via MIA or go straight through into LGW on the 777. BDA is a good departure point as well for F.

As for AA vs. BA, the service on AA is worlds better, but the AA seat is correspondingly worlds better than the BA. If your sole intention is to sleep, take AA via MIA. The route is almost always Flagship Suites, you'll have acres of room and while the service isn't the best, it's good enough.