I personally refrain from making a booking on 3rd party websites / OTA's and book directly with the airline, so as to hold them entirely responsible for any lapse / IrrOps.

JP Miles to be redeemed for a AA codeshare on a BA operated flight => a receipe for disaster.
You are bound to make it very easy for one of those 3 airlines to blame the other 2.

None-the-less, Good Luck..! ^