If your fill water is high in TA or if you have a lot of water evaporation, then the evaporation and refill can increase the TA level as well (everything in the fill water gets added to the pool). Since your water is very hard, the TA may be high in the fill water as well.

Anyway, it is a known fact that increased aeration will result in faster pH rise. The solution is to get the TA as low as possible since that reduces the rate of this carbon dioxide outgassing. However, an SWG has other ways of raising the pH such as having chlorine gas that doesn't fully dissolve in the water so outgasses from the (usually closest) return instead. And I already mentioned how you are adding way too much acid if it's full-strength since it would lower the pH too much. That just causes faster see-sawing. Also, half a gallon of full-strength Muriatic Acid in 13,000 gallons should lower the TA by 19 ppm while half-strength would lower the TA by about 9 ppm. I'm telling you, something just isn't right about your situation and I suspect some information (maybe test results) are off.

Look on the label of the Leslie's acid that you have and tell us what is lists for the ingredients. Is it this stuff? If so, then that should be 31.45% hydrochloric acid (i.e. full-strength Muriatic Acid).

What kind of test kit are you using? You aren't using test strips by any chance, are you? Or are the numbers from a pool store?