No no, if you just use Stuhl it works perfectly in German. This word has exactly the double meaning that is necessary for the pun, it’s not even a stretch. I’d have more trouble finding an appropriate verb for the translation.

A Schemel is really a very small seat, no taller than knee-height. It’s something you sit on to milk a cow for example.

And a Hocker is a kind of chair with usually neither arm nor foot-rest of any height. I’d say it’s pretty much a perfect match for the English stool. You could call a Schemel a kind of Hocker, but a Hocker can also be as tall as a normal chair or even taller at counter height, in which case we’d call it a Barhocker.