Hedera helix 'Erecta', if it grows there.

Wow the list is endless...but I would start with saying that I'd plant something evergreen as the birch is deciduous. You could either plant a shade-tolerant conifer (Cryptomeria j. 'Knaptonensis' would be kind of cool, as the new foliage is white and would match the white on the birch trunks. Or there are lots of broad-leaved evergreen shrubs - Abelia 'Silver Anniversary' or 'Confetti' have lovely variegated leaves and brown stems, again matching the birch nicely. There are also a lot of Berberis with colored foliage, some of which are evergreen, but I don't know how they perform in your zone.

Here is a link that might be useful: Form and Foliage

I don't think the erecta will work here, but i could be wrong. Otherwise, i like the idea. I also love the suggestions from formandfoliage. I will love into each of those options and sincerely appreciate your comments.

birch are notoriously surface rooted.. and your facts dont really tell us how big your trees are ...

you might want to TRY to dig a hole there and see what is going on in the soil .. before you spend lots of time making a plan .. and buying stuff ...

if you can not dig a hole.. or there are so many roots that anything you will plant there will be out-competed for everything it needs .. then it might not be the best place to plant ...

not saying it cant be done.. just saying you might want to investigate the soil first ...

and if its really bad .... it will be a different list of what might thrive there ...

ken