I would definitely stick to Provence and the Southern Alpes. Plan to spend several days in each of a few little towns and just explore from there.

I'm not sure about rental cars with racks, but I am sure you could do it - the bigger question for me would be how to store your bike boxes. When we did the bike tour, the tour company kept our bike box. The next year when we rented an RV, the RV rental company would have kept our bike boxes for us, but we ended up not bringing bikes. I would ask the car rental company in advance if they would store your bike boxes for you.

I know plenty of people (including my husband) who've taken bikes on the TGV from Charles de Gaulle to another city. They make it seem harder to do than it really is. Helps if your bike box is moderately sized.