For what you are doing, you really need one car in two bays, not two cars in one bay.
You want to do your body work on the good frame. Doing it on a twisted frame will result in a twisted body. I would put the good body car in the bay. Strip it down until you would have to cut metal to get anything else off it. Like Rob said, rent a POD or a storage unit. Buy a shed and put it in your back yard. Bum space from friends. I brought home a pickup load of pallets discarded at work, tore them apart, and put a floor in the attic of my garage. Stored most of a 57 Chevy there.
Do the same thing on the donor car (outside I would guess). Then, once you are sure everything is disconnected, raise the bodies and block them. Swap the frames. With floor pan and quarter work, I wouldn't prep the frame at this point. Get the body work done and then pull the frame back out. If you have everything out of and off the car body, and it is structurally sound, and you support it well, you won't do any damage. Check your rocker panels inside and out very carefully. If they have rust damage, you may end up with two halves of a Monte. If in doubt, weld 1" square tubing to the inside rear quarter running to and welded to the kick panel. One top and one bottom with a diagonal in between them. My 41 actually had pretty bad rockers, but I didn't put braces in and it lifted fine.