Need a much more detailed/close up photo to determine.......and without any flowers, even that may not be enough :-)) However, if you need to replace the trellis, there is no reason you can't cut the clematis back now. At most, you will just be losing a seasons flowers. As long as the primary basal stem is not huge, you could just do a hard prune now although I would be inclined to just cut back as far as you need to to disentangle it from the trellis so that can be replaced.

A closer photo is needed since I can't be sure that it's a clematis from the photo. An alternative to replacing the entire trellis if only the bottom has rotted and you'd rather not replace would be to get a couple of 5' lengths of rebar and a package of black zip ties, pound in the rebar a foot or so alongside the uprights of the trellis, and then use several zip ties to fasten the trellis to the rebar supports.

Looking at the growth habit, I guess it is Sweet Autumn Clematis. My SAC grows just like this. Heavy pruning is required to keep this in check.

Thanks for the replies! Here is a closer picture, I hope it helps!

I have two of these same ones, and there is a lot of dead branches underneath and behind the green ones. I am hoping that I can do a hard prune after they flower. They are quickly reaching over my 6' fence.

I don't think it's sweet autumn clematis (C. terniflora) since the leaves don't look quite right to me, but it is a large one. If it blooms in the summer, you can prune down annually over the winter or in early spring before the leaves start growing again.

Some clematis will rebloom if pruned after blooming if the season is long enough.