From a position of understanding, although no expert, this comes under Human Trafficking Legislation. I imagine that in this case the fact it was made via AA so theres a chance that data didn't have the chance to tie up with that held by BA there is and never will be any exemption to the protection stance that any airline takes. You need to understand that there are substantial financial, not to mention the more important consumer repercussions and most importantly the protection of those being sold into slavery of any kind, of any airline not doing the checks they are required to carry out ( in the UK at least I can't speak for elsewhere) legally to establish the validity of you travelling with a minor who is not your child.

I 100% understand how in a legitimate case like your own you would feel this is stupid and to a degree insulting. All you are doing is taking away a loved family member on a holiday that maybe they could never have via mom & dad for whatever reason( rhetorical statement no need to answer as it's private business). To give you an idea of just how strict the legislation in the UK is regarding this subject I took one of my nephews away a couple of years ago. We flew from our home airport of LGW and my partner is a senior member of the security team there. We had a letter formatted correctly giving us permission to take him out of the UK from both his parents however we got pulled, were separated ( myself and my partner) and questioned by the police to just make sure. If it can happen to somebody who knows the police on a personal level at the airport we were flying from & meets with the security services at least once per shift then, imho, it's reasonable to appreciate that you are given some extra hoops to jump through. It's not a personal slight it's to comply with legislation and again imho VERY important legislation. I wont deny for a second that "we need to talk to the 9yo travelling with you" is, to be frank, moronic as no boxes would be checked as said child at that age could easily be coerced to answering the way they were told to by a trafficker & wouldn't protect any airline from prosecution for exactly that reason.

BA being a British airline has to comply with any legislation of their home nation regardless of where pax board or alight( if it was found that somebody was trafficked then for penalties to be imposed they wouldn't need to have ever stepped foot in the UK BA would still be liable for allowing it to happen). Taking into account the scale of human trafficking nowadays it's an unfortunate but necessary step to ensure that any child moving from one country to another is safeguarded properly.