I suggest you go to the library and read our host's Versailles instructions. It's practical as a half-day from Paris, eliminating the luggage worry and car rental timing. As CJean noted, most U.S. departures are early enough from CDG to be risky by car the same day. If you are content to miss attractions along the way anyway (Le Havre, Rouen, Giverny, also a possible half-day trip; Chantilly) you could spend the last night in a hotel in downtown Paris and get a little more than three nights in that great city. RER train or cab to CDG in the morning. Three hours is barely enough for Economy Class check-in lines.
Another hint from our host, "Assume you'll return." Although car is a good way to get from the Loire to Bayeux, it's also a dreary three hours or more on the highway, with a few sights each 25 minutes (each way) off the highway. You won't have time to stop. It is true that a car is very useful for D-Day sights, although you didn't specifically mention that.
As is often pointed out here, you would benefit from starting to think of your trip as "nights" instead of "days". Since the first and last days are necessarily devalued, you don't really have enough time to do three regions of France justice. You can make the best decisions if you plot out specific interests, like certain specific Chateaux, and what you want to do in Normandy. Time of year makes a difference in full hotels and traffic.