My top recommendation for a family would be a campground on the Atlantic coast south of Bordeaux. They don't open very early in spring or stay open very late in fall. I stayed at one in early June (before the big summer crowds arrived). They have pools, restaurants, laundry, cabins to rent (or tent sights) and are walking distance to the beach. Mine had a small lake and you could rent paddle boats. You can rent bikes and ride the bike trails to lakes, beaches or the big sand dunes. I picked one about 25 km south of Arcechon, not far from Bordeaux. Nothing was expensive. There's a ton of similar places that all offer very similar experiences. The reason I'd recommend it is you will meet European families, it's a very family oriented activity. I had nice conversations with an older couple from Rotterdam and a young couple from New Zealand who were camped next to me. This is something that, once you arrive, you could live without a car while staying there. Everyone could rent bikes and it's as safe as it gets.
Burgundy is beautiful, lots of tiny medieval towns dotting rolling hills covered with farms and forests. Some areas are vineyards but they don't overdo it. You're more likely to see grazing cattle or horses than vineyard after vineyard. I visited Burgundy in early May and felt it was too early to be there, it was quite cold and most things were closed. I wouldn't suggest it before at least late May. The tiny towns probably won't have much to keep your kids interested without a car. With a car, you can visit some fairly interesting historical sights. I think they'd enjoy Guedelon, the open air museum where they are building a castle using 13th century plans and techniques. Kids can ask questions of the different artisans and find out about making fabric, tile, stone cutting, etc. Most are French but many speak English and/or are English artisans who come to volunteer for part of the year. There is the battleground site (arguably) between Julius Caesar and Vercingetorix that gave Rome control of Gaul for good. I was amazed that both sides wore very similar armor - with a couple key exceptions. There are a lot of really old churches and towns, if you're kids like medieval stuff. Bibracte is a former Gallic town and museum, see what it was like when the Romans showed up. Obviously wine and food are big draws for the region.
Provence? Only in the fall. The spring gets Mistral winds, sometimes for days on end, that will seem like you're in a hurricane zone. I went in late September and it was still very warm. A friend went in the spring and all he remembered was the wind.
The Dordogne area is really pretty too and there are activities, generally you need a car. The kids would likely love the caves with Troglodyte art. There are canoe rentals all around the Dordogne and Lot River Valleys and some of the nicest castles. Most Dordogne towns were rebuilt after the 100 years war, when this area was a front between the English and French.
I liked the Languedoc area a little better. It's also beautifully forested, like Burgundy and Dordogne, but the towns are older (not razed liked Dordogne), more spread out and there are castle ruins on hilltops you can hike to. I was there in 2018, you just hike up and wander around. Other castles are restored and charge admission fees. Languedoc doesn't get as many visitors so prices are really reasonable. Locals told me the Spanish side of the Pyranees is sunnier, but I found the French side to be much prettier. A car to get around is also necessary here (as in Burgundy, Provence and Dordogne/Lot).