Laura, most clematis will not skip a beat if they are trimmed back. They will put out new foliage and may even reflower depending on the length of your growing season. If I were going to cut my clematis back in your zone I would do it by midsummer at latest so that it will have time to get up and get growing and establish new vines that will flower next season.

When you do trim your clematis back, I doubt you will have to trim it down all the way to the ground. Cut the top half back and then see if you can't just unweave some of the branches out of the previous trellis and lay them on the ground. Then insert your new trellis, and weave those branches back up into the new trellis. You may break a few branches doing this but those that do break you can trim back to the point where they break and not all the way to the ground. Regardless of how you do it, I would leave at least two or three nodes on each stem with active growth on the vines. I would be leary of cutting a type II all the way to the ground not knowing if you had planted it deeper in the ground than it was in its original pot since doing so buries dormant nodes that will sprout new vines.