For the big one, get a long enough pole or even a straight tree or shrub branch stripped of leaves and side branches. Secure it tightly to the short stake you already have first, then loosely tie the entire plant to this new support.
WATER WELL. Then gently lay the whole thing on its side on a sturdy cardboard, supporting the BRANCH with something to keep level and avoid crushing the plant/leaves.
You should be able to pull off the pot after pushing on its sides to loosen, and maybe with some help. Use the support stakes for handling as much as possible. Loosen potbound roots gently but don't worry about it too much.
Prep the (much) bigger pot. Lift the rootball with the cardboard as support (obviously with corrugated cardboard the lines should be vertical so you could enfold the rootball) - someone else should be supporting the upper part with the stake. Slide it in. Fill the rest of the container with potting mix and water well.
...something like that. Well near enough. I don't worry about the roots so much unless it's a long-lived perennial or a houseplant. Ha. Yeah rainbow is right -- the new pot should be nearly ready before removing the plant from old pot, minimize root exposure.
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