Yes, that kind of double dipping is fine!
Just a couple of things to keep in mind ...
AA Elite restaurants don't all give BA "basic" (non-elite) dining credit - heck, some won't give any BA credit at all in a few instances. Check your lists before you dine. Never assume a restaurant gives credit in all of your airline plans just because it does in one.
To maximize your overall earnings while working towards BA status, you might think of splitting tabs during this AA promo period so that at least $15 of each dine goes on your AA card, and the balance towards BA status. In your case it might confuse servers as they are identical-appearing cards with different embossed numbers, but worth a try.
Good Luck!