I would consider 2, 2-week stays. I've done one week stays in several places in France.
That said, a place with a pool near the Dordogne River somewhere in the vicinity of Beynac and st cyprien would be where I would most want to spend an entire month. There's a lot of great things to do, amazing food, a relatively predictable climate, no mosquitoes and river beaches to swim at (in addition to the pool) and night markets to hang with people. To me, this is head over heels my top choice.
If I were to break it up, my next choices for 2 week stays would be northern Brittany, Catalan country, and the Alps. But I think I would get bored spending an entire month there.
One week in Provence was plenty for my family. The mosquitoes in Provence can be a drag. Not a big deal if you stay in a city but the country house we rented with a pool was swarming with them in the evening, so we spent more time inside than expected.
The Loire is wonderful but one month would be too long for me.
I've not been to Normandy and I can't stand the area around and including Nice - just my opinion.
One more note for what it's worth I love Aqualand water park. There are 5 or 6 of them in France. Water slides are the best.
Happy planning.
Matt