Thanks to everyone for all the suggestions.
@balso, yes, we can only go in July due to husband’s job/daughter’s summer vacation. Not ideal, I know. He really wants to visit that area. Thank you so much for the suggestion! I will definitely look into that town. I think we are going to avoid being directly in St Tropez for now. Since we are pretty intent on no car, I’m leaning stay in Nice and we can ferry or bus over to StT for a day trip or something. I know Nice might not be as “glam” but I think it’s a good home base with at least variable transportation option.
As someone mentioned, we are now planning on Paris > Bordeaux > back to Paris (for a couple nights to eat/rest/laundry) > train/bus to Nice > bus to Milan which I realized is only 4 hours away via bus, and fly out of Milan. I’d personally push a little more to visit other places, but my kid is 9 and doesn’t like to move around too much, so the plan is 4-5 nights in each town (aside from 2nd Paris visit).