To put your mind at rest I have worked for Flybe and British Airways and at no stage have had a British Passport. I have a 'right to abode' through marriage visa stamped in a foriegn passport. In either interview was never asked specifically if I had a UK passport, just asked if I have the right to live and work in the UK. Certainly with FlyBe you will never go to a destination where a British passport is specifically required. Go for it.
Cheers
