As mentioned above for a BX6B if there is MileSAAver space showing at aa.com it should be bookable through Business Extra. I book a lot through Business Extra and this has mostly always been true, with one weird exception (I wanted to fly with a connection from SNA to LHR via DFW as the only first class space I could find was DFW-LHR, and SNA-DFW had space and aa.com displayed the itinerary, but Business Extra tried to tell me they don't have space if ticketed together ... it made absolutely no sense). Anyway, the overall key with your BX6B is it must be on AA metal. So your options into Europe are limited. If you want to take the 777-300 to LHR, your award to Europe will end there. Any BA flights from LHR-Europe will need to be booked separate. If you want to get to Rome or Zurich, you need to find a AA metal departure landing in that city.

An important note is as of tomorrow this is [supposedly] changing. Starting tomorrow, when Business Extra merges with BA's On Business in North America, Business Extra awards will be available for travel on BA and Iberia metal. What I was told is it's going to be a completely different award tho. So an award on British in first class from the United States to Europe will be 20,000 points. There will be no saver awards on BA. The representative I spoke seemed to think this will be Anytime space (or close to it). However, this probably doesn't help you with your existing BX6B. But there's a chance BA space will be available tomorrow so it may not hurt trying back in a couple days to see how this develops ...