I don't know if you will see this (already in Cannes?) but here it goes:
It's raining and very cold at the moment, which is unusual. I remember having lunch on the beach with just a t-shirt in previous years around this time.
When it rains, there is nothing to do - we don't have soft play areas, nor much indoor activities of any kind for babies/children because weather is pretty much always quite pleasant. There are several large shopping centers you can spend a few hours in - Cap 3000 between Nice and Cannes (~20 min drive), or Metropole in Monaco (~45 min drive). I hear it is snowing inland, so driving away from the sea is probably not a good idea these days.
If sun reappears soon, stay in Cannes. There are quite a few cafes/restaurants on the beach (yes, they are open), and I think your 10 month old DD would enjoy playing with the sand. At the west end of the beach where there is Palais des Festivals (where stars walk the red carpet during Cannes Film Festival), there are some kiddie attractions like a merry go round, which might interest your DD.
Sales start tomorrow (Wednesday) so shopping could be interesting as well. Haute Couture is on La Croisette (seaside road) but best shopping imho is on parallel Rue d'Antibes.
Enjoy