In order to board a BA flight to the UK with your Bahraini passport, it needs to have an entry clearance in it. Since you don't have one, the answer is you should use your British passport. You should make the reservations with that passport so that your name will sync up to your travel document. Michael Hampton's answer addresses how to handle Bahraini exit controls.
For your second question about where the information goes, it goes to the Border Force whenever you enter or exit the UK on a registered passenger carrier (or more to the point - whenever they ask for it). The information you give gets combined with other information like your credit card details and the number of pieces of baggage you checked and the list goes on and on.
It's available to the Border Force before you arrive and after you leave. If they think it's interesting there are agreements with other countries to let them know. I don't know the case for Bahrain, but it's likely they follow the same model.
The UK's solution for dual nationals who wanted to conceal their British nationality has been traditionally to issue the person a Right of Abode certificate. This certificate does not mention nationality at all but allows the person to travel to the UK without a British passport. You may want to see if you qualify for a Right of Abode certificate.