The OP might also consider doing his initial award search at aa.com, which is capable of displaying award-seat availability on AA, AY, BA, and IB (among others). It is better than ba.com at displaying connecting award flights.
Any AY, BA, or IB award flights displayed on aa.com should be bookable with Avios. With AA now moving to dynamic-award pricing for AA flights at the MileSAAver level, I'm not sure if all AA MileSAAver award flights will continue to be available with Avios.