Hi! New member here with hopefully quick question about lounge access. We're considering an AUS-LHR-OSL trip, and maybe springing for business class because it will be for our anniversary. All segments of the trip are British Airways-operated flights. I am wondering whether it makes a difference, for Galleries lounge access in LHR, whether the trip is booked through BA or through American. The reason why we'd be considering booking through American is that BA does not participate in TSA's Known Traveller program but American does, so (I presume) we would not get TSA Pre-check if we booked through BA, but likely would get it if we booked through American. Does lounge access depend on whether you are booked in the appropriate class on a BA flight, or whether you actually booked said flight through BA?

Thanks!