Chlorine Bleach & pH

[waterbear]

pH is rising, though, is that due to the high alkalinity?

Yep. the higher the TA the faster pH rises (because of faster outgassing of carbon dioxide)
The lower you put the pH the faster the pH rises (because of faster outgassing of carbon dioxide)
The more you aerate (move) the water (by water features, spillovers, fountains, splashing kids, negative edged, SWCGs, etc.) the faster the pH rises (because of faster outgassing of carbon dioxide)