For example, put some display tank water in a pail, net the fish, put into pail, pour water and fish into hospital tank?
This sounds like a good idea, iyou should also acclimate to the hospital tank
Never put water from the hospital tank into your main aquarium, and I like very large nets when catching fish. Usually this makes catching them easier, which is less stressful on the fish. Prior to actually netting the fish (sometimes you got to) I try to heard it into a plastic container, so the fish can be removed with as little physical trauma as possible.
You should have nets and equipmet, just for the hospital tank