So if I have this right, you'll be somewhere in Provence with a car on Sept. 10 and no plans for the next two weeks, but you want to visit Dordogne, Loire, Brittany, and Normandy -- or some of those areas, including Normandy -- in that time. And I'm going to assume that your flight home is from Paris, though you didn't say so.
So, using Via Michelin to find routes, and assuming you start in Arles, it's 6-8 hours (depending on route) to Sarlat, a good base for the Dordogne. Then 5-6 hours from Sarlat to Amboise, a good base for the Loire. Then 4-6 hours from Amboise to Bayeux, a good base for Normandy. Finally 3-5 hours from Bayeux to CDG airport. In other words, four full driving days -- the Michelin tool doesn't assume any stops, and tends to be optimistic about traffic and drive times. But that leaves you 11 days, more or less, for those three areas. I think that's enough time to experience all three adequately, if not in depth. Adding Brittany would be a stretch, but you could substitute it for Dordogne or Loire, perhaps with an extra driving day. Of course, your starting point and choice of bases might differ from the above.
Not knowing more about your interests (history, art, food, wine, stuff for kids, whatever) it's hard to make recommendations. You must have a guidebook -- what draws you to each area? All are superb destinations, worth more time than you have. You might do better to pick just one besides Normandy, and save the other for a future trip. If so, the Loire would be easier to reach from Paris at some future time, so that might be the tie-breaker.
Hope this helps, have a good trip.
EDIT -- I forgot that your subsequent message mentioned MSM. In that case you might want to spend time in Brittany with MSM as a stopover on the way to Normandy, and save Dordogne or Loire, or even both, for a future trip.