I cant remember the correct torque setting, I'm sure I saw it on a danfoss bfp pump YouTube video.
The drive key should be solid in its groove with no play, if there was it could cause your issue and eventually shear completely and you would have no pressure or ignition. A photo would of been nice but I'm not suggesting open up again. A dodgy pump coupling could also cause the same.
The burner could draw in flue gases if the flue seals had failed or burner flange gasket was faulty, or anywhere else that could allow products of combustion to escape inside the casing. You could detect these products with a FGA yes.
Just out of curiosity have you ever dipped the tank for traces of water with detection paste? I'm sure this isn't the problem due to what I've seen but thought I'd ask.
With regards to the quote for changing the pump I'm not so sure if that will help now based on what you've said today but without seeing the gear set, drive key or coupling I cant say for sure.