Misters help in DRY heat, but are useless or even worse than useless in high humidity.
I've had good luck with dampening PART of a shady section of ground -- not to make mud, just to get it a little moist. But let the hens pick whether to dig into the dry ground or the damp ground.
Some people don't mind that kind of intensive cleaning. :)
I cover some of the pros and cons in my Deep Bedding article:
https://www.backyardchickens.com/articles/using-deep-bedding-in-a-small-coop.76343/
Dirt alone without litter just accumulates poop and doesn't give it anything to react with.
I'm a great fan of the Deep Litter method for the run -- a moist, active composting system where the organic litter reacts with the poop to neutralize it.
Welcome to BYC. Have you read my article?
https://www.backyardchickens.com/articles/hot-climate-chicken-housing-and-care.77263/
A tub of dirt wouldn't help them cool off, though they might like a dustbath.
It's the ground itself that's cooler once they dig themselves in under the surface...