Southern France Itinerary - Please Advise!

My suggestion would be to forget about the beach club so you'll have more time to do things you can only do in France, but I'm not a beach person even when I'm not incurring the costs of a European trip.

ViaMichelin.com will suggest driving routes and calculate estimated driving times for you. I have no experience driving in Europe. I used the regional trains and buses as necessary in that part of France in 2017. While a car is very helpful--for some destinations essential--in Provence, the usual recommendation is to use public transportation along the Riviera. Traffic down near the coast can be really bad, and parking is challenging and expensive.

Monaco and Menton are served by trains. Eze has bus service, I think with an intermediate stop in Monaco. It might also be an option to take the train to Eze-sur-Mer (not the part of Eze you want to see; you want Eze-le-Village) and get a taxi up the hill--but I've never done that.

Rick has a guidebook just for that part of Southern France. I think it would help you a lot. He'll warn you that most of the beaches are rocky (requiring beach shoes) and are packed with rather expensive lounge chairs. I'm sure he makes suggestions about the best beach options.

Would you be interested in Villa Rothschild? I really enjoyed leisurely photographing the gardens and having an espresso & sweet on the terrac after viewing the villa.

Thank you Jean for reminding me. Yes, we will spend a morning at the Villa Rothschild. Right up my alley!

It might it may be too much in one day with Menton, Monaco and Eze together. Now, Eze is tiny and won't take much time, so it is definitely possible. One option is to do Eze on your free day and combine it with Cap Ferrat. If you don't mind a steep walk, you can walk down from the upper village and catch the bus to the Villa Ephrussi. Or you can take a bus or cab town.

You can take a beach day and go to Villefranche sur mer or Antibes (or Juan Les Pins). That can help break up the beach day, after you hit peak sun. The beaches to the west old town Antibes (in Cap d'Antibes into Gulf Juan) are sandy. There are beach clubs but it is not all the space is taken by beach clubs. The VSM beach is lovely but large grain sand, with one beach club. Really only in Cannes do you see the long line of beach clubs along La Croisette (and the less famous beaches are way more open).

You also need some time in Nice, during the daylight hours, to go through the old town, the markets, Castle Hill and any museums you would want to see. .

Your current list of destinations do not require a car. You'll find trains, trams and buses to be sufficient. I only rent cars for excursions into the countryside and even user Ubers at times (within Nice, not an option in some places).