I don't know if this would help with a chinchilla, but when it gets really hot here - hot and muggy, (bleah) we freeze a plastic bottle of water, and put it near where our rabbit relaxes/snoozes during the hottest parts of the day. It creates a little cool zone, and of course collects condensation which she can lick off for a nice cool sip of water. It's cheap, (they're my leftover Fruit2-O bottles), and by having two bottles in rotation, there's always a cool one waiting.

We don't, in Massachuestts, have the constant heat/humidity, but we get stretches of it that feel like forever!