Well if you used Airpots you wouldn't have to worry about pulling them out of their pots, you could just unravel/unrap the pot. There would still be some work depending on how you attached the Screen and to what, but it would be easier than with normal pots. If your screen was made out of a textile instead of a metal the plans could hold it up while you transplant. its what I plan on doing.
I'll use nets instead of metal/screens, so when I transplant all I do is unwrap the airpot from around the root mass and then gently put it into its next air pot (the plant should be advanced enough it can hold the textile screen for as long as it takes to get it secured again. The cool thing too about these is if the airpots are differen't enough in size, you could fill the larger one with soil mostly and then place the other pot in and unwrap it with it already in the new pot (then take first air pot out and fill with soil)