Generally it is not a good idea to move peony so soon, especially with only 1 stem but it could be done. Just do not expect it to perform for 2-3 years. Try to move it with as much original soil as possible and I do not think that you need to plant it in the pot with new soil as long as moving time around a week or so.

I agree, moving them in the fall won't kill them. As long as you can wait until they start to go dormant they'll survive the move. If you can afford to buy new roots to plant at your new place you'll get flowers quicker, though. I'd buy from one of the online sellers like Adelmans, Hollingsworths, Hidden Springs or such and those will possibly give you blooms next spring--the ones you have now will probably be at least 2-3 years after being transplanted 2 times in 1 year, even being dormant. Unless you're incredibly lucky you'll lose the small roots they managed to put out this year.