As I understand it. The airlines have nothing to do with who gets TSA Precheck or not. IN the beginning, many US based airlines offered the opportunity to apply for the trusted traveler program (like Global Entry -which I have) to their most valuable frequent fliers as a way to jump start the whole TSA Precheck thing. Yes, when you complete your travelers info on the your airline website there is a place to enter your trusted traveler - known traveler number and that number should also be listed on your profiles for any airline you have "status" with. BA might not "offer" Precheck but IF you have "status" with a partner airline and you list that information on the BA site AND you are a known traveler - you very well should see the TSA Precheck on your boarding pass (with of course the random denial for whatever reason).
As mentioned above TSA has stopped just randomly giving TSA Precheck to folks not enrolled in the trusted traveler program as the traveling public has had time to enroll or not enroll and the program has accomplished the goal of allow frequent travelers to get through the TSA lines quicker.