I have been diagnosed with reflux and I don't get the typical heartburn either. Mine is all in my throat. Plus, I have the cough you describe with asthma like symptoms and lots of mucous. I take 20 mgs of Nexium twice a day and it helps about 70 - 90 %, depending upon how conscientious I am with what I eat, how I eat, when I eat and how much I eat. I am having the scope on Friday and guessing that the GI won't find much inflammation at the bottom of my esoph either since I don't have the burn in my chest. In regards to the acid getting into my lungs . . . I found that if I do any heavy exercise, the acid will get into my lungs and I'll get that wheezy cough. So, I have taken up walking (strolling) instead and have the lung thing pretty much under control. If I start coughing, though, then I think it snowballs and I can't stop and then my lungs burn. So, when I start to cough, I'll pop a few Tums and they seem to help long enough to get the coughing under control so that I don't get the wheezy cough.
Anyway . . . you can still have GERD even though you don't get typical heartburn. If your doc doesn't believe you, check out some of the websites on GERD to confirm your symptoms then take them to your doc. I like my doc, but she sure doesn't know everything, especially when it comes to the finer details of the reflux.
Good luck to you.