When you do the actual fragging I would place the entire coral in a small tank/ bucket and do the cutting in there submerged along with however you choose to attach it to the rock.
During the process the coral may slime. By using the seperate water it will reduce the amount of waste released into main. Once done let it sit for awhile and then just place it back in the main.
I would try to limit any exposure to air at all other than for the actual transfer. Some corals are more sensitive than others.