I usually fly into BAH or DMM via FRA or LHR. The problem with your proposal is that I have to enter Bahrain using my Saudi passport (Saudi and Bahrain share intel and stuff and Saudis have the right of abode there). Plus, if I fly into Bahrain with a US passport and leave Bahrain via the Causeway using my Saudi passport, they'll know something is amiss.
As I said, there are so many Saudi-Americans who travel routinely despite have their names spelled differently. I'm just not sure how they deal with it in terms of buying a plane ticket.