Here are some that I grow that might look nice:
Huldine is white.
Venosa Violacea is bright, deep purple on the edges and white in the petal centers ( which would look great with the rose, but not sure about the fence, depending on its color).
Mrs. Robert Brydon has oodles of tiny whitish-lavender flowers but would need some help climbing or could wander around the feet of the rose. It's quite vigorous, so it would need planting near but not on your rose unless it too is quite vigorous.
I have what I'm pretty sure is Will Goodwin (can't currently find the tag,) a lovely light lavender-blue with a white center, but I have heard from others that type 2s don't do as well in southern areas. Maybe someone from farther south can weigh in on this.

The opposite of peach on the color wheel is light blue, so any of the paler blue toned clematis would look great and contrast against the darker fence - The President, MultiBlue, Cezanne, Filigree, etc. One of my favorites is called Teksa, which means 'denim' in Estonian and is exactly the color of a pair of well-used blue jeans.

In my old garden, I had a planting area that was focused on orange/peach and purples and I grew peachy colored roses like Just Joey and Apricot Nectar. I combined these with very dark purple clems like Etoile Violet and Gypsy Queen for the contrast -- it was a very successful combination.

Prince Charles would be nice

Your help is very much appreciated. I'll Google these so I can see what they all look like. I imagine they are all beautiful. It's fun to look them all up. ~ Thanks again to each of you.

Gardengal, by the way...you mentioned Just Joey. I saw that one in a photo once and ever since, I've always want one of those. One of these days...lol