I just travelled on an airmiles flight on BA - not sure if AA redemption allows that transatlectic though. Availability in premium economy and business is usually OK.

Also, try Rome, Milan, Prague, Warsaw as they have daily flights to SOF. Warsaw with LOT could be interesting - last time I did that there was only 1 other person in the business cabin from JFK and they have beds also.

If you google sofia airport you can see who flies there daily from europe.