ginger50,

Those flights used to be operated by BMED, a BA franchise.
Unfortunately, I was in the same situation as Beirut was part of the same network.
If you book the BD flights with BA flight number, it will be around £150 more.
I would recommend that you:
-either book the flights directly with bmi (www.flybmi.com) for much cheaper
-or go with BA to Dubai / Abu Dhabi / Bahrain then connect to Emirates / Etihad Airways / Gulf Air respectively, all of which are top quality luxury airlines (I know these airlines very well as I am Lebanese flying all over the Middle East on those airlines).
I would not recommend, however, BA to Doha then Qatar Airways because BA's flight to Doha first stops in Bahrain.

Hope this helps,
BA6501

PS: by the way, BA6501 was actually the flight number from London to Beirut when BMED existed