May I suggest that you contact Nick Laing at
this company.
Nick was one of the first people allowed to take tourists into Bhutan. In fact, my father introduced Nick to the country, as the recently retired/abdicated king, Jigme Wangchuk, was a boy at Heatherdown prep school (now closed), of which my father was the headmaster/owner, and the king very kindly invited my parents and my older brother and sister for a visit in 1969. They were amongst the first westerners to visit this amazing mountain kingdom.
I am not certain that Nick will be able to advise on specific airlines as such, but he certainly will know the best ways to travel there and which airlines may well show interest.
Hope this helps?