As asked, "Can BA see the TYO-SFO segment?" If not, the itinerary is phantom and the discussion is all for naught.

Stating Avios doesn't show the award flights isn't helpful as Avios isn't an airline loyalty program. If the TYO-SFO segment cannot be found on BA, the TYO-SFO cannot be found on AA as a rule. AA may have further married segment logic, though. The "logic" on BA and AA ceased to match some months ago.