Let me state this a different way:
You plan on registering your 2nd credit card -- of the account where your original card has already been registered with AA - with BA. You can do that, no problem. On your next restaurant visit, you will have a choice: get 10 AA miles/dollar using card #1, or 3 BA miles/dollar using card #2 (which will also count as the first of your required 12 dines for "elite" BA Rewards Network status).
Here is what I believe you feel will occur, and which I can assure you will not ...
You use the 2nd BA-linked card to get 3x BA miles plus 10x AA miles because the primary card is linked to AA?
Nope. No Way. Forget it.
One cannot "double dip" between airlines on the exact same transaction; Rewards Network does not recognize any "link" whatsoever between the two cards. Your dine with card #2 will post as 3 BA miles/dollar; there will be no posting at AA at all. If you contact Rewards Network about this, they will (correctly) reply that you were eligible for BA miles with that credit card, which you've received. Period.