How does my itinerary look?

Finding a 7 passenger van is probably not too difficult. A 7 passenger van with space for 7, PLUS the luggage, may be the challenge. And if you need AT the pool of available vans shrinks.

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This is a motion-packed itinerary, you'll be moving your family on seven separate days, a third of all the days you have in France. Each time you move you're packing, checking out, either getting a train or driving, finding a new hotel, checking in, and unpacking. So most of those days are spent just moving yourselves, with a lot of scenery but little time for much else. And you're covering pretty much the whole length of France, a country the size of Texas.

This website will help you find rail options all over Europe, including France. I'd be surprised if you could find a train from Provence to the Loire on Day 16 without having to go into Paris and change stations, a major hassle. But without a city in Provence to use as a starting point I didn't try to research this further. Maybe there's a convenient way by train, but my guess is you'll want to drive.

You have only one full day in the Loire (Day 17), a big area with lots to see, and you'll be tired from the previous day's long journey. But you have three full days in Normandy (Days 19-21) after a shorter journey there. Both areas are great, and you may have strong reasons for wanting to spend that much more time in Normandy, but you might consider a more even balance. Bayeux is a good base for Normandy, you have more choices for the Loire but Amboise would be a good one (Blois might be another).

On Day 22, might as well drive right to CDG and get a hotel there, especially if your flight home is in the morning. The drive isn't difficult, certainly easier than into Paris itself. If time allows you could go back into the city for your last evening, or even overnight if the flight is late enough the next day.