Well, there are some actual recommendations for kiddie pools here. And yes, I realize it’s a pool chemical manufacturer’s website, but it’s a good illustration I think of the procedures that you have to go through in order to practice chemical sanitation on even a small kiddie wading pool: you gotta test, add chemical, and test, and test again. There’s no easy way to do it. Here’s an .edu website with basically the same information: It’s a LOT of work. If you’re gonna keep it going, you’re going to have to be out there testing it three times a day.
Or you can just dump it every day or two, like everybody else. We had kiddie wading pools of various sizes in our backyard from 1987 to until just a couple of years ago, one of which was a big hard-sided 30" deep pool that Daddy liked to float around in, and dumping them every couple of days, even the big pool, didn’t harm the grass in the long term, as we always moved the pool’s location from week to week. And we never noticed that the water had trouble sinking in. It went somewhere, since our back yard isn’t concrete.
The main problem was in hoisting/folding up the side of those injection-molded Wal-Mart pools in order to dump, but there’s a knack to it, you discover. You get it part of the way, you let water flow out, and then it’s lighter, and you lift it up more, and more water flows out, until finally it’s empty and light enough for you to flip it over.
I notice that the pool you’ve picked out really doesn’t hold that much water; the inside dimensions are considerably smaller than the outside dimensions. So it shouldn’t be that much of a PITA to empty and refill. It’s designed to be a toddler frolicking place, not a place to “swim”, i.e. submerge. They’re supposed to play on the slide and just generally fool around, not swim, so it doesn’t hold that much water.
Also, with such a small volume of water, if you’re going the pool chemical route, and you do the math wrong, or mis-measure, and accidentally get too much pool chemical in there, there’s less volume to buffer your mistakes, and an increased chance of nasty burning eyes, or worse, due to too much pool chemical.
Really, just empty it every day, especially since it’s a two-year-old who’s going to be using it, and who is statistically more likely to pee into it without realizing you’re not supposed to do that.