By making the item you are drifting into homebrew. Even if rubber doesn't exist you could speak to your gm about homebrewing it in
You might want to look into the bombchucker.
actually if you talking about the bomb class feature, it specifically says you can
"A bombchucker doesn’t get in the way of preparing and throwing a bomb created with the bomb class feature."
Most of what we call rubber is actually some sort of complex synthetic. However, real rubber from a rubber tree is not terribly complex to make. Sargava would be a likely source in golarion.
I think if you want something like a slingshot for flavor, that would be fine, I would model it after the flask thrower in terms of game stats.
Basically the flask thrower is like a Lacrosse stick, while a bomb chucker is like a Jai Alai cesta.
that binds a hand and makes it so you can't on the spot make potions.
as the hand wielding the bombchucker can’t be used for any other purpose
The thing in your video definitely occupies the hand. I'd say it would need a longer-than-free action to pull it off. Might require both hands, too.