Here's what I would do:

I would dig up the peony NOW and pot it up. I would remove all flower buds as they appear so the plant does not spend its energy on producing flower and seed. Keep it in partial to full sun in the pot throughout the summer being sure to keep it properly watered. When fall comes around again, divide up the plant into 3-5 eyed sections and plant wherever you like.

Hm, how far out of the ground are your peonies right now? You might be able to get away with digging, dividing, and transplanting now, if it's not too far along. Though, I would still not let it flower this year to conserve the plant's energy.

And now for the experts to critique...

Thanks, Lactiflora. I figure it's at least worth a try. The peonies are about a foot out of the ground here right now, leafing but not budding yet.

-Maeve