Have each person bring say 5 stones. A judge tosses very low quality out. Remaining stones are randomly assigned. Perhaps problem solved.

I'm coming to realise that people will compete at anything. And cheat at anything.

The stones having to come from the island messes up my thoughts on this but you could have “natural” erosion of the rock in a circle if you made a synthetic riverbed that flowed water through very quickly, while dumping sand in. Like a weirder rock tumbler. Would also take a very long time to get appropriate rocks and I’m not sure how you keep them from going into spheres, but anyway.