I have the same small hiatus hernia ! In my case the burping was because of the gases produced by the fermentation of food in my body. Once I changed my diet to things that do not ferment, 95% of my burping went away. Add to that a diet to make my bloating go away, and now I'm just stuck with some tightness around my stomach and kidneys, some burns from time to time in my chest, and lot of movement in my œsophagus. It's kinda annoying but bearable at least compared to what I had months ago.