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.
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 (plane or ship). 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.