I don't know why not - they're different things entirely. Of course you need to credit your BA flights (if any) to your AAdvantage account, but buying the ticket through BA shouldn't make any difference.

The way the Plat challenge works, you get a 100% mileage bonus on the flight during which you exceed a cumulative total of 10,000 elite qualifying points, and on all successive eligible flights. With your FAT-DFW-CDG-LHR-LAX-FAT itinerary, that milestone would be reached during the LHR-LAX flight, so including cabin bonus miles (50% for first class, 25% for business class) AND the 2015 special bonuses for non-elites which again is different from the special transatlantic bonus, you would probably end up with something like 50K miles in the bank per person by the end of the trip counting the Plat, TATL, class of service, and "special" bonuses. Enough for a FC round trip in the US, or a one way business award to Europe or Asia. Not too shabby.