1. While not mandatory, AA would certainly refund tickets in this situation. But, you are stuck with BA as the ticketing carrier.
2. I would not be quite so blase about what a FA will do. The policy expressly includes age 2+ and was changed to eliminate all exemptions (along with UA, B6 & WN). Both B6 and WN have had high-profile denied boardings (or removals) at age 3. No reason to expect any difference from AA.
Nonetheless, I would call back and, if denied, ask for a supervisor/manager. If unsusccessful, I would pay the fees, and then write in asking BA to consider refunding or crediting the fees as a gesture in light of the fact that the operating carrier imposed new requirements after you had purchased the tickets. These are simply fees. It is not as though BA does not recover the value it transferred to AA to carry you.
I would not attempt to fly and risk having the family left at the airport or going through a miserable experience.