You must call AA to book. It is possible to search for availability using the British Airways website.
Be aware that LATAM has very limited availability for its partners. I have been trying to book a trip to Argentina and I have never seen a single award seat available. You may be more successful to Ecuador, but availability will be spotty for one, let alone two. You may want to focus on flying American to Ecuador. That availability will be OK, especially midweek.