Bart, I have moved many princess trees, but as very small plants, usually first or second year. They have a tap root, and I don't know what size they can reach before you'd be murdering the Princess in trying to move her. Here, they are considered invasive (which is why I have moved several, lol), but they're so pretty I can't bring myself to just trash them. I just keep watch for seedlings (easy to do since they can reach 3-4 feet in their first year) and diligently remove the excess. I do know that they must be moved with as much of the tap root as possible, AND they need a root ball. They can NOT be bare-rooted. Every single one that I didn't manage to keep a decent amount of dirt with the roots, it just died. Speaking of roots, I think paulownia can be started from root cuttings, since nearly every one I've moved...stayed, lol. At least it came back from the remaining roots.
Ken is correct in saying they should be moved while dormant, but that time is coming up soon. Just make sure you do it before the end of March, as that's about the time they start to break dormancy (at least here). They bloom about Mother's Day, or before in my garden.
As an aside, did you know that in Britain they are often grown as foliage plants? If a mature specimen is cut to the ground each Spring, the new growth can sport leaves up to two to three feet across!
Q