I have been having the exact same issue as you have (Pretty consistently for the past 5 months) and find relief with certain strategies. The issue to my understanding is that reflux material such as acid and enzymes and aspirated material (same stuff) originally make you hyper sensitive. This will cause as well as exacerbate asthma symptoms. The reason its so hard to pinpoint the source is because both air irritants and your lpr/gerd play equal roles in the breathing difficulty.
This past week I have managed to almost entirely get rid of my shortness of breath. Not sure if this is just a temporary fluke but here's how I am doing it.
I have been experimenting with "HCL + Pepsin" (one tablet before each meal), Allegra allergy 24 hour (One per day), chewing my food slowly and thoroughly to reduce burping, vitamin B12, and Vitamin D3 (there's allot of people that say this works well). Also, before I workout I have a puff of my inhaler. I also take a couple tablets of Gaviscon before working out, sleeping, or if and when I feel my stomach starting to "reflux".
This has been fine tuned after trying about 40 different supplements/combinations over time. It seems that all of the other stuff may not have had much of an effect.