Follow up after my recent experience:
Booked flights with AA (status less with them) using my BA FF# (OW Sapphire), as I know I won't be reaching gold I thought about crediting the miles to AA instead.
AA's website won't let me change the FF# whose main perk online is the free main cabin extra seat selection, so I decided to give it a try during check-in at the airport. I spoke to an agent who managed to change the FF# to AA, and told me I had to present my BA card to get priority boarding and AC access.
His advice worked fine throughout the trip, my BA card did the job nicely.