I've had several barium swallows, and they aren't very sensitive to small things, but seem to catch big issues instead. In my experience there's more of a risk for false negatives, rather than false positives. You can be having mild reflux, but if your LES is not wide open, it's not likely to show anything.
As far as what you're described, I would guess that you're getting plenty of reflux, but I assume you're on PPI's. If that's the case, then the PPI's are protecting your esophagus and keeping the esophagitis at bay. (That was generally my situation. I did get esophagitis from time to time, but my esophagus was always healthy during endoscopies and biopsies. Thankfully.)
I'm no expert, but that would be my take.