As someone who also suffered from atypical symptoms, I feel your pain. Doctors are really only comfortable recommending surgery for those who have an open LES and food and acid refluxing freely into the esophagus (and mouth). They know that these patients are sure to experience an improvement in symptoms once there is a repair and the wrap becomes a backup valve for the non-functional LES.
To a certain extent, I can understand their hesitation. This surgery is a life-changer. Even when it does exactly what is required and the patient is thrilled with the results, there will always be changes in the GI tract that will need to be accepted and adjusted to. While they won't have the horrible reflux any more, there may be swallowing issues, or intermittent dumping, etc. A person must be very positive and realize that these inconveniences are worth the effort, given their improvement in quality of life.
I faced the frustration you are feeling for years, as I visited and revisited GI doctors--changed doctors, and had multitudes of tests. My overriding symptom was asthma. I had reflux, but it was not the unbearable type that is experienced by those whose LES is wide open. That said, I had enough "silent reflux" to create a serious health issue. I even spent a week in the hospital with GERD-induced asthma that was out of control.
I could not get a GI doc to take my symptoms seriously. Finally, after four years of horrible lungs and damaging doses of steroids, at long last, my GI doc suggested I talk to a surgeon. Nobody was thrilled about doing the procedure, as there is no guarantee it will solve atypical problems.
My surgery definitely gave me a better quality of life, and helped my lungs. That said, I still get some reflux, and at times it's enough to affect my lungs. I've been to a different surgeon (we moved) and asked if I needed a redo. After testing, he said that my wrap was intact and perfect, and he wouldn't touch it. There is no way to get reflux to zero, without creating eating issues.
So...I suggest you go to another GI doc and pursue your concerns. Be persistent. Understand, though, that the surgery isn't as perfect a solution for atypical symptoms. If you are willing to take the chance that it will help you (I did), and embrace the quirks you are left post surgery (it's a major revision to your upper GI), go for the surgery.
Good luck with finding a good GI doc,