I don't know much about bile reflux. I looked into it breifly when I thought my gall bladder was causing problems, but I ended up having a nissen for a hiatial hernia instead, so my experience is obviously limited.
BUT- I was told that when I was taking carafate for my reflux, that it was good for the bile reflux as well. You may want to ask the doctor if carafate will help you. Its a very benign drug, although its a pain to take because you have to take it an hour before, or two hours after eating or taking medications.