juliabagg,
Although it may be difficult to find a place at this point in time, you could try checking gites-de-france for rental of a house or apartment to stay in whatever town you settle on in Provence. They operate a bit like AirBandB, but have been around longer, are very reliable and have a variety of places to stay, from rooms at a real B and B to large houses. Southern France covers a large area...the Riviera (Nice) and the Luberon, Drome, Provence Bouche de Rhone. When deciding where to narrow down our trips, I usually get an old-fashionbed paper map, spread it out, and see where things are in relation to one another. It gives me a better idea of distances between places so I can eliminate outliers, get a better idea of time to drive between places, and what I can reasonably find time to visit. (e.g. Pont de Gard is close to Nimes, but not too far from other charming towns as well.) What you finally decide may help you determine the mode of transport from Milan. Suki has made some very good points about this, as have the other posters.
And Lola brought up a point which I, and maybe those of us who are past the small child ages, forget...car seats! Rules are very strict in France about secure seating for the little ones. And in Italy as well, I would guess. Something to consider, especially because larger cars like minivans are difficult to drive in many smaller villages. Parking is problematic also. (I speak from experience. In 20012 we had an American sized minivan because we had a relative who was elderly and mobility impaired. It was no fun for my poor husband!)
I hope you are able to get your plans settled and that you have a wonderful trip! Bonne chance!