Yep, copper will kill off anything, whether its on a sponge or rock. If you ended up dosing copper though you would be doing a good amount of water changes anyways, you have to keep the copper at a specific level every single day.
In my mind if I needed a qt for treatment I would most likely do water changes daily anyway. I wouldnt wanna risk any live rock for sure, even if you dried it out the copper would have been leeched into the rock and could slowly release back into the water at a future date.
Depending on what you would be treating for, hyposalinity would probably be the best bet for ich. In that case you wouldnt have to worry about copper leeching.
Keep in mind I have been pretty lucky and have never had to set up a qt for treatment but thats what I would do if need be.