Q. Clematis suggestion?
Clematis vines do well in full sun with the roots shaded. You idea of growing a ground cover under the vine is a good idea since it will shade the clematis.
Meadowbrook Farms is a local nusery where you can buy your plants. but they are closing for the season at the end of September. I suggest using google to local mail order sources of your plants.
There are many good cultivars of clematis plants. THey bloom at different times in the season so make sure you are picking one that corresponds to your desires.
You might try ajuga or pachysandra, myrtle, sedum, and mondo grasses are others.