If I were you and the planner, I would set up this trip like a tour. Arrive in Paris and see all of Paris, whatever you think those number of days are needed. I would suggest a minimum of 5-7 days. With one of those days to Versailles. Then decide on west of Paris or south of Paris. I wouldn't try to see both, it will be a lot more travel and wasted time moving about packing-unpacking. Keep in mind, there are many day trips you can make from Paris. Chartres, Reims, Loire Valley chateaus from the city of Tours, etc.
If you decide to go west. I would suggest basing in Bayeaux. From there you can schedule tours of Normandy WWII and a day tour to Mont Michel. In Bayeaux don't miss the Bayeaux Tapestry. It is a great region to visit.
If you want to travel south, you can base in Avignon. Take the TGV in about 2:45 and explore the region from there. Many nice day trip cities to see such as Arles, Nimes, Pont du Gard, etc. You can TGV back to Paris the day before returning to the US.
Another alternative is Lyon and Annecy. Lyon is a great city in France and not as far as Avignon and Annecy is a beautiful city on Lake Annecy at the foot of the Alps which is a great representation of a French Village, IMO. These two are a great combination as well.
Good luck with your planning. Enjoy France!!!
Also, keep in mind the time of year. If it is late fall or winter, Normandy won't be very enjoyable. If it is summer, south will be hotter. The lavender fields you speak of are at their peak during the months of mid June and mid July.