Depending on how dry it is, you may be up against the same problem with hostas. They are not as water needy as impatiens, but they do much better with ample moisture than without. You might be best trying good ol' 'Undulata Albomarginata'. Not real exciting, but a strong grower...maybe a little bigger than what you had in mind, though.

What about sinking a pot in the ground to keep the hedge roots out? I have a similar problem under a tree in my yard and have thought about just potting the hosta. Never done this before, so I am not sure what to expect.

I'm in the same zone and area as you are. Hosta would likely be a good idea. Mine is planted in all sorts of shady and dry spots here, and because I live in a forest, tree roots are a constant.
We do not water our gardens. Many plants don't even get watered into their new homes. (I'm very lazy.) But they are always planted with copious amounts of compost and I sidedress with compost every few years. Hosta will lay down deep enough roots that they can access water in all but the driest spots. They will be easier and less thirsty than impatiens. However, if you do water, hosta will grow bigger and faster.
Buy good hosta. Undulata marginata tends to be slug bait. So buy a hosta with slug resistance.
I would suggest Minuteman, Night Before Christmas, or Twilight. All grow very well here and all are slug resistant.

All good suggestions above. Also, you might want to look at Golden Tiara, it's a small-medium hosta, about 36"W x 16"H. Lovely and very hardy, a bit of a spreader.

Have you considered laying a soaker hose down there? Both your shrubs and the hostas would appreciate it. After all, we want the hostas to be all that they can be!
Peggy

I am a researcher compiling the history of Carl Borgstrom's landscape architectural work. I discovered a reference by you indicating that your garden was designed by Borgstrom. I would appreciate knowing more details, especially where it is and if any design records remain.

Alexander