Lots of interesting answers. I'm not an expert, but I could guess an EK 77W is too big for BFS which was why I was surprised that EK flies it from NCL.
I would have thought QR with their 788 could make BFS-DOH profitable. There isn't a huge difference between a 788 and 321 in terms of seat numbers. That brings me to my next point. QR has the A321NEO on order which should have the range to operate BFS-DOH, maybe thats the real option?
I suppose the really feasible option is Turkish Airlines, they could definitely do it on the 321/739. They'd get people wanting to holiday in Turkey as well as people connecting long haul.
In terms of J class, I reckon most NI ex-pats working in the ME could probably afford a J class seat, or else QR staff could use them on standby tickets; ok the company makes very little money, but better than nothing.