I don't know what would be the best place with a car, but others here will be able to help if you create a new thread with the key elements in the title--something like "Base with Car for Short Visit to Dordogne". We have quite a lot of posters who've traveled in that area with a car; it's just that I'm not one of them.
I believe there have been positive comments about staying in or near Beynac, which is near Sarlat-le-Canada, a popular destination for its large, twice-a-week market. But the key thing would be to do a bit of research and decide on a list of villages you hope to see, then pick a place convenient to them. You don't have a lot of time (a week would be better), so you'll probably want to limit how far east you go as well as picking an optimal location central to your targets.