From your description, it seems you want to be able to completely remove the netting from the hat, not just stash it out of the way. Is this correct?
I like the idea, and wandering around Joanns you will probably get lots of idea. I would say you will want to attach straight to the foam of the hat.
I'm not sure how thick the foam on the Chillba is, but one option could be to use small metals snaps that Joanns sells. You can get about 8 of them in a pack, which should be enough for the project. It would involve only hand sewing one half of the snap to the hat and doing the same for the other half on the netting. I would probably put a small reinforcement patch on the netting as well. In the front you could position the snaps farther away from your face, as if you had a bill to lay the netting over.