Hi Janni,
Your Clematis belongs to group 3. You can choose any other type 3 Clematis to plant next to RC. Make a search on the Clematis on the web web site for this group and for the colors you want. There are hundreds of possibilities.
As for Peonies, I do not think you have to consider them when choosing the color of your Clematis: Peonies usually bloom much earlier than group III Clematis.

Alina, thanks for a quick reply! I am familiar with COTW, but I just think it is difficult to choose a clematis to match my RC based on their photographs. Plus when I google a specific cultivar, the images I get show huge variations in color. I was hoping someone here, who had perhaps seen several clematises in bloom, could offer specifc suggestions to help me narrow down the search.

Janni,
the thing is - they do have variations in color! The same cultivar can look differently even planted within the same garden. That being said, there are some varieties that performed well for many people. Some of those that were recommended here many times (that satisfy your color requirements): Betty Corning, Polish Spirit, Petit Faucon, Venosa Violacea. Basically, all Viticella Clematis.
If you are that concerned about color combination, go with paler varieties. The Rouge Cardinal has very bold color, so combination with lighter colors will always work.

Petit Faucon is an integrifolia and much smaller than Rouge Cardinal.

Here is my trellis with Rouge Cardinal, Venosa Violacea and Price Charles.

{{gwi:583059}}