Please don't give up your pets!
I lived in the Keys for several years. I know what your climate is like. Everything grows and prospers, including mites.... This actually isn't much different in south GA, where I live.
You can manage the mites, really! Treatments work on the living mites but not on the eggs. You must treat again after two weeks, to kill the mites that hatch from the eggs. And you must dust or spray both the chickens and the whole coop. And the run, if you keep them in a run.
Sevin dust from Wal Mart or the hardware/garden/feed store works. Pyrethrin/permithrin poultry dust from the feed store works. Adams flea and tick spray from the vet's office works.
Many people feel DE works as a preventative, but that is a whole other discussion, and it must be food grade, which many cannot buy except online. But you can buy the other stuff I mentioned locally, I am sure. Actually, I control mites mostly with DE, but I must treat now and then.
They will get mites again, because the mites come from wild birds. People say, if you have chickens, you have mites. So it is just a matter of staying on top of them. A few mites will not kill the chickens and will not infest you. Just dust or spray regularly, and please continue to enjoy your pets. They will need to be treated for mites wherever they live!