Clematis should have no trouble climbing a wooden lattice trellis.

Good type III bluish colored clematis for your garden include Perle d'Azur, Prince Charles, Victoria, and Wisely.

I find that they could use a little help on trellises with wider slats (1 1/2" or so). I weave fishing line throughout those types of trellises to make it extra easy for clinging.

I have some wooden trellises that the slats are 1/2"- 3/4" and those are just right IMO.

I have great luck with Prince Charles, which is a light blue, Emilia Plater is very similar to PC. Blue Boy integrefolia is also a good blue, but a bit deeper and more purplish.

I can't speak to zone as I am slightly warmer

Once the side branches on the vines get long enough they can cling and wrap the ends of themselves around just about anything provided there is enough friction for them to grasp. Blue Boy is a nice clematis but is not a clinger although it might periodically throw out petioles that can wrap themselves around things. I would not count on it though.

I have to manually weave my Clems on my cedar lattice and on my bird netting trellises too. I wish they did a better job on their own.

Mine seem to climb very well on their own regardless of what they are grown on. Bird netting is very small so am surprised you have to help anything climb on it BorS.