I have a friend who has a house in the Pyrenees that she is just about to start renting out. The area sounds to be exactly what you are looking for. Her children are mad keen cyclists and they all love it there. She drives down from Brittany several times a year - think it's about 8 hours? I'm sure she could tell you more if it sounds suitable?
We do a few Eurocamp sites in the alps and Jura region - can't remember whether these are campsite or the town names but lac de chalain in Jura and bourg d'oisans are both good areas for what you describe - bourg links directly up to les deux alpes ski resort which has loads of summer activities going on
There are lovely towns pretty much everywhere in France. If you want hills, the Jura is nice. I have also always heard that the Vosges mountains are lovely and if you were based there you could visit Strasbourg/Mulhouse etc that are supposed to be really interesting and even hop over the border into Germany.
Otherwise, Burgundy and Loire valley very scenic and full of interesting places to visit, but pretty flat.
That won't be the worst weekend. Usually "Black Saturday" is the first Saturday in August when I am travelling and maybe the Saturday after. You could get people travelling back that weekend but hopefully in the opposite direction to you (unless you will be travelling East to West).