If a Bahraini citizen, who lives in Bahrain and holds a Canadian passport, wants to travel to Europe using his Canadian one; will the airline accept presenting his the Canadian althought he will check at the Bahraini passports control point with his Bahraini passport? Which passport shall he use at the time of booking the ticket?
It's fine, at least as far as the airline is concerned. This situation is very common and airlines are quite used to dealing with it, plus all they care about is whether you will be let into your destination.
- Book your ticket with the name in your Canadian passport.
- Show the Canadian passport when checking in for your flight.
- Show your Bahraini passport at Immigration.
Now the one catch is that Bahrain apparently does not recognize dual citizenship, so if the "Canadian" name on your boarding pass differs significantly from your "Bahraini" name, or they notice that your Bahraini passport doesn't have an EU visa, Immigration may start asking awkward questions. The traditional dodge for this is to fly via a third country with two separate tickets, eg. Bahrain->UAE (with your Bahraini name) and then UAE->EU (with your Canadian name).