France in 15 days

6 days Paris (one day trip to Versailles)

4 days Normandy (stay in Bayeux)

Remainder Provence/Riviera.

I agree with your husband. You can rent apartments this way, and get to know the area better. In Paris there are lots of apartments on VRBO, but I also like the Hotel Eber Mars.

I basically agree with Donna's plan, except that there is no single place from which one can tour both the Riviera and what is generally known as Provence (generally, Rhone valley and Luberon plus Marseilles area). They are too far apart.

On the bright side, there is plenty to see and do in Provence--you won't be bored! Fly open jaw to maximize your time on the ground, and take the TGV between Paris and Avignon. (That means two car rentals, since you will need wheels in the provinces.)

The larger issue is this. Rick's rule-of-thumb list is not bad, but is based on a super-fast, super-efficient pace that is really only feasible on a tour, at least for a sustained period.

If you are very focused, experience, and nimble you could do it on his terms, but even then I doubt you would feel it was the optimal trip you could have had in 15 days.

And the requirement for only no more than 3 base destinations quite rules it out.

Donna's itinerary has knockout destinations. However, expect crowds in Normandy around the D-Day anniversary. Reservations should be the order of the day there.

Assuming that your days don't include travel time to and from France and within France, I would opt for Paris/Versailles, Loire, and Carcassonne. You'll get the beauty of Paris, the rural loveliness and chateaux of the Loire, and the very different area around Carcassonne down in the south, which is one of my favorite regions of France. There are people who will say that you can see easily see Carcassonne in a day and they are correct but the surrounding area is stunningly beautiful with lots to see - you could spend weeks! I am a big fan of going to a region or city and plunking down for a while in an apartment. Get to know a few areas well, rather than try to cover many. Spend 4 or 5 days in each area and you'll have a more restful, satisfying trip!

One thing to remember is that this year in Normandy will be the 70th anniversary of D-Day so would not plan to be there then. You will probably not even be able to get reservations there as you are going May to early June.

If possible, try to go a month earlier. You can see now that air fares are cheaper at the end of April than the start of June. That would also avoid some of the D-Day congestion. Sadly, the 70th anniversary carries some urgency since it will be the last chance for so many vets entering very old age. A beaches tour aimed at Juno Beach and the tank battles around Caen will teach you a lot about what the Canadian trooops endured. Either Caen or Bayeux could be your base. Normandy is big and has other attractions too.
Otherwise, less is more. Paris/Normandy/Avignon would be a full two weeks, allowing enough time to get a real feel for the places you stay, and with good train connections. Or, rather, consider the order in reverse. I prefer flying from Toronto into Marseille and then moving north to go home from Paris. Departuretimes to Toronto are more convenient out of the big city.