BA works great if you learn its quirks. Search each individual flight, plan to book one-ways as round trips often do not show up. Unfortunately there are no coach seats this week, but there are JFK-LAX first class for 37500 Avios, and LAX-JFK business class for 25K. I did find one routing through STL that showed up on AA but not BA - you should be able to book anything from AA along those lines over the phone.

Start here, log in, search for one-ways in each class: http://www.britishairways.com/travel...n_type=STD_RED