Hi Cumulus-nimbus. Peonies can be kept in very large containers in more temperate climates. They aren't advised in the colder climes because the extreme freeze/thaw cycles they'd go through in pots would kill them.

We have members of the forum that propagate peonies, either by seed or cuttings, that keep their peonies in 3 gallon or larger pots until they're judged hardy enough for growing outdoors, sounded like 2-3 years from the posts they made. I've also seen posts from people in zones 6 and up that say they keep their peonies in pots year round. For the warmer climates it's probably easier to get them into a chilled state for the winter and shadier area in the heat of the summer with pots.

As far as planting medium you should be good with any quality potting soil, just make sure it's well draining. You certainly don't want to let them dry out too much between waterings but it'll kill them even faster to be waterlogged. You might have to experiment a bit with adding some sand or cactus mix to your potting soil to get that sweet spot for keeping the soil moist rather than saturated with watering at a frequency you can support.

Sorry my advice isn't first hand on the pots, every one I tried here in zone 5 bordering on zone 4 died. I envy those who can grow in pots so have paid attention to their posts, the advice I offered was from what I gleaned from them.

Good luck with your seedlings and make sure you post some pictures as they grow!