You can't mould clay balls, or at least it's not practical. In industrial ceramics, you'd make a cast two part slip mould but then you'd get flash and it's hard to get them out intact.
In the fishing industry you'd use a radial boilie roller (grooved rollers), or a Rollapro, but I really doubt that's practical for you and you'll not likely find one the right size for slingshots. BTW, the radial boilie machines don't make perfectly shaped spheres because they are designed for speed not perfection.
Marbles are made on a polished helical roller with roller with gaps smaller than your ball. These are more accurate than radial rollers.
Ball bearings are made by cold-heading and then grinding under rill-plates.
I think the best way for you is to do it the traditional way where you make a consistently flat sheet of clay, score it into equal sized squares and hand roll them into shape. PS, if you have a cheap source of local labour, I am sure you can get all you need.