Possibly a honeysuckle? Some of the hybrids are not invasive, & they like having their roots cool. I have a clematis with very limited sun below 4 ft, & it is doing fine. I'd say that any reasonably fast growing vine will reach the sun in short order, but stay thin in the dense shade.

My being in zone 7A makes my recommendations for zones 6 iffy.
I have seen hops vines and dutchman's pipe vines used, and they are quite hardy, but not evergreen.
Climbing Hydrangea is a classic, and a vine called Schizophragma 'moonlight' is grand. But neither is evergreen.
My favorite evergreens, here are Gelsemium sempervirens and Clematis armandi, but I do not know how they would do in your zone, and the former is overwhelming.
American Wisteria sorts, like the clone called 'Blue Moon' have none of the faults of the Japanese or Chinese invasives.
At any rate, whatever your choice, vines seem to do better planted a little away from the structure. About a foot or so.