Some of the untrackable buttons came early! The seller split the shipment into two—one with the red and one with the other three. This one was supposed to come after the single button package, so, bonus. I can get three quarters of the way there.
As it turns out, the stool comes apart and opens flat.
The button shafts measured 1". At my brother's suggestion, I bought a Forstner bit and drilled slowly.
Drilling through the stool was easier than I expected, and went a lot better than I imagined it would. You can probably see the X bracing on the inside of the stool, which is the thickest part (7mm).
It took a while to drill all the way through, but the hole is nice and clean. Well, clean enough.
aaaaaand it fits! The buttons are nice and clicky, but not too loud.