Clematis armandii if it will grow in your zone.

That's a tough question! I think your best choice would be a vine that normally takes full sun, but "needs some shade in warmer climates." I agree you should look into clematis -- a tough Viticella variety or native (western white clematis?).

A passiflora could also work. I recommend browsing this site: https://www.brushwoodnursery.com/

Clematis armandii grows in bursts of quickly developing, long shoots making diligent management necessary when it is being attempted to keep it within a circumscribed space such as part of a built structure. Also like perhaps most climbing plants it will soon pile up on top of a support if that support is much lower than the typical height of the climber (15'- 20' is indicated by published descriptions in this case - but I have seen evergreen clematis having grown up wires to near the top of a multistory building in USDA 8) unless regular pruning and training is undertaken to prevent this. Finally this plant retains its old dead leaves so these need to be kept snipped off in order to maintain an attractive appearance.

Otherwise your depiction worded

I only got 3 on each side last year! Even tho the roses reached the top where those canes actually will get sun

makes me think you may be misreading what happened with inadequate sun exposure not being the issue. With some critical involved points being what kind of roses, what if any pruning has occurred and just how much sun exposure they got.

the bridal vine stephanotis