I had a difficult-to-diagnose cough for a few years that turned out to be asthma, though I’ve never had an asthma “attack.” The diagnosis required a specialized test called a methacholine challenge. Finding the right steroid inhaler did the trick for me, but it took a lot of trial and error (and fighting with the insurance company once per year for approval). You also mentioned reflux, and whether or not that has anything to do with your pouch it can certainly cause a chronic cough. It would be very useful to find a way to get the reflux under control for a while (e.g. with acid controlling medication) to see if your cough gets better.
Good luck!