Cap Ferrat has also a nice indoor pool. For only 2 nights I can´t see it´s getting boring. If you wonna go outside for excample Villa Ephrussi de Rothschild is near by and a great spot, also Villa Kérylos. Don´t know it but very near by is Villa Santo Sospir. If the weather is nice you can also use the walk along the beach. I would stay with FS.
Regarding beach club: Restaurant is only for lunch anyway and the pool is nice but nothing you can spend the whole day. Also don´t expect beach weather. The ocean is definitively not swimable at this time and if you wonna enjoy the sun you can also do it in the garden.