You'll likely want to stay away from AAdvantage, because you'll only get 25% mileage (and 25% EQM towards status) on your discount economy BA flights.

As for BA, you won't come close to BAEC Silver with the amount of flying you've said you'll be doing, nor will you do so with QF, though you'll get 100% mileage on each program for the itinerary you've given above.

Happy flying, and all the best finding the right program.