If you've booked already when you open up seat selection in MMB what cabin is that showing you? If you haven't booked then try searching for the flights on BA.com rather than the app as that will get you to the seat selection page without having to complete the booking.

AA Trans Con's don't have a PE cabin yet. It's First, Business and Economy until the new fleet arrives later this year. I'm therefore not surprised to see a long haul PE ticket book into B class economy on the domestic leg where PE doesn't exist (though it would normally be marked up as Economy B in the page view you posted). The JFK-LHR flight I would however expect to be booking into a PE fare class (E,T or W) rather than B.