Just booked an AA sAAver seat via BA (phone agent) even though nothing was showing on BA.com or in her system, but I could see T class availability on EF. She called the OW desk who advised her to try and book a seat on the flight the pax wants, if it accepts then there is reward space available, and they can proceed with the booking.

So if you have EF access, look for T or U availability, and if it shows tell the BA agent to try and book it regardless of what their system is showing as available (or not available).

Hope this helps!

Pilot37