Ich is parasitic. It's commonly found in the environment. IF the fish gets stressed, it's immune system can't fight what's out there... cold temp i'd say will definitely kill the fish's immune system making them susceptible to disease.
I've had ich in my tank before and I didn't remove the fish. i treated with ich medication, bumped the heat up...let the parasites fall off, and then did big water changes until everything was good to go.
And just b/c you love something doesn't give you the right to mistreat it.
It's like saying: hey, I love my son, but I'll just let him freeze.
Reefing is pretty damn expensive to keep up with. If you're worried about the $ it takes on upkeep and making sure the tank and inhabitants are in optimal conditions, I'd suggest looking into a different hobby.
It amazes me how people mistreat coral & fish... "oh it's just a fish." Regardless of how small an animal is...it's an animal.. it's a pet... coral are alive... your tank is alive.... treat it and respect it like you would had it been another human being or living entity that you might respect or revere more.