As the atlatl, one can make a "stone-thrower", using a cannonball (please ignore the klicky hat).
Both the atlatl and stone-thrower can be fully functional, and if you have a klicky where the shoulder joint is a bit loose, a fast fall of the arm, perhaps via a thread tied to it, will throw the dart or cannonball.
I do not think something like a "cannonball atlatl" has ever existed, but I think that with two mutilated "large soup spoons", we can make a sling as those used by Balearic mercenaries of Hannibal, in his war against Rome. One of the spoons will have the handle cutted off, and will be glued to the other spoon, retaining the handle, at both ends.