I've never divided a bleeding heart before, but I know with my peonies, I have found the best time is in the fall, after blooming (and most growing) has ended. I generally dig a large circle around the plant I want to split, then grab the garden hose and gently remove all the soil from the root system.

After getting the rots mostly clean, I try to find areas of the roots which are only loosely connected and split them there...

In my experience, splitting peonies in the spring has resulted in almost no blooms that year. However, when I split them in the fall, the next spring, while not as profuse as if I hadn't split them, I still had blooms...

Good luck!

I've moved bleeding hearts at any time they needed to be moved. This might be a little early, but if your ground is not frozen, I would give it a try. Make sure that you firm in the plant you move it. One advantage to moving it now would be that it should fill out early in the season.

For blooms, I think it is better to move peonies in the fall.
I did pot some up one fall and let them freeze in a protected place, so I could take them with me when I move in January. I held them at the new house until spring them planted them, and they are still growing 25 years later. I think it would be hard to divide a peony without damaging it if it is frozen. I might wait until spring and sacrifice the blooms for this year.

Divide both as soon as you see new growth. Don't plant your peonies too deep or they won't flower.

Here is a link that might be useful: My Garden Travels

So fall for the peonies and now for the bleeding heart. Thank you all!

Jim