Make sure you get the export licenses from Bahrain first, otherwise you'll have to drive/ship the car back to Bahrain to get them before you can register your car in Qatar. And make sure the car can and will be registered in Qatar, I.e. There might be some laws regarding how old the car can be, I think 5 years max. You have to check that.
I agree with Iron, pay 200BD and let a company take it for you. Some of them will do all the paper work for you, and have it delivered at your door in Qatar. With all the beaurocracy involved I would think that this is a much better deal, since they are fully aware of the processes involved.