Hi Mike, I'm drifting into an uncomfortable zone, as I'm trying to second guess your clinicians, which, on the internet, is not always a good idea.
You can sometimes get symptoms of reflux cough, without the burning feeling of reflux. I'm just wondering how your AME has ruled this out, as it's sometimes difficult to tease out in the history and examination.
I have known some clinicians to trial an "empirical" course (ie a trial and error approach), of a "PPI" (a drug to reduce acid.) Again, not trying to second guess your team.
I think it's reasonable to ask for a consult with a respiratory physician.