Have you tried pricing the itinerary with the morning flights? That wouldn't be considered a stopover so shouldn't impact the fare. For >24 hours, the system should, in theory, find the lower of either (1) a fare that allows a stopover (either free or for a fee), or (2) a constructed fare by combining BDA-JFK (in business) plus JFK-LHR (in first), if the fare rules allow combinations (which they may not).

That said, I do see in ExpertFlyer a ANN4AZN1 fare on BDA-LHR (travel date 11/15) for $2200 (which probably excludes carrier surcharge). It does allow one free stopover. It's also an MPM-based fare (4,140 here--longer than BDA-JFK-LHR), with ticketed point deductions for various routings, and that's beyond my knowledge base!

Maybe try calling AA?