Actually, the ants aren't required for the peonies to bloom. They're just attracted by the nector (or so says a professional gardener friend of mine). I usually deal with the ants by cutting the buds/bouquets when they are just about to open, put them into a vase with water and then leave them outside for the night. When I bring them in the next morning, the bugs all seem to be gone.
I've always heard that transplanting peonies should be done in the fall. I would think that it could be done at other times, but you may find that the plant produces fewer blooms for the next year or so as a result. Heck, I just planted two peony plants that I bought from a nursery two weeks ago. That's not a whole lot different than transplanting. I planted another one last year about this time, and it's already producing blooms this year.