If you haven't had recurrent oral herpes lesions, probably you do not have oral HSV-2. It's possible, but not likely; and if so, asymptomatic oral shedding is much less common than for the genital area. Despite you neurologist's opinion, I doubt your burning sensations are not due to herpes. (Did your neurologist specifically come up with that explanation, or only after you prodded him for an explanation?)
Presumably your opening statement means you have had symptomatic recurrent genital herpes lesions. If you have oral herpes, the outbreaks of the mouth will be essentially the same: blister-like lesions that appear, typically on or near the lips (and never inside the mouth), turn into open sores, then heal with scabs over 7-10 days. The description of your oral symptoms doesn't sound like HSV.
If your partner develops herpes, I have no way of knowing whether he acquired it from you or someone else. Perhaps the first step is for him to be tested to see if he has HSV-2; and, if not, for the two of you to take appropriate precautions to protect him. Also, if your HSV-2 infection was never confirmed by lab testing, you probably should have a blood test yourself.
Best wishes-- HHH, MD
Doctor is it possible to have both herpes 1&2 orally?