Have you thought about putting it in a large pot in a sunny spot instead?

I suppose I could, I've just had the experience that nothing grows as well in a pot as it will in the ground. Would it do well in there for a long period of time?

You say you bought it in a little box for five dollars, so I would guess it is a pretty small plant. For this year I think I would pot it up in a one gallon pot and focus on growing a plant with enough root to compete with the tree, I think it will. Al

Winter shade won't matter, the peony should be dormant. I would be a little more concerned that you don't get cold enough, but it is worth a try. Make sure you don't plant it deep. The eyes need to be less than an inch (this link says 2") under ground. In the south you can plant peonies in partial shade. More sun than shade is preferable.

I have peonies planted under my crepe myrtle trees, but further on the edge than right up close to the trunk.

Here is a link that might be useful: Gardeners.com