In reply to Tall Clare: BKB Jr isn't so little anymore (yep time does go quickly)but we are used to little visitors.
TO the OP:
THe 2 places for climbing that I would really recomend (though there are others) would be
Roche Sabots: very short easy walk in, kiddy circuit and general small boulders to climb on, large enough areas of sand to play in and there is shade if it's hot.
Canche aux Merciers: very short easy walk in, kiddy circuit and general small boulders to climb on, very large area of sand to play in and there is some shade if it's hot.
I would not recomend Cul de Chein because there is almost total lack of shade but if it is overcast then littleone might like it, walk in is longer but doable for a 2 year old.
What I would be thinking is how are you going to take all the stuff a 2 year old needs to the rocks. We used 3 wheeler as added benefit was it gave somewhere for littleone to have afternoon sleep and bouldering pad can be slung over the handles.
When littleone could walk in we still carried in a very light weight maclaren buggy so there was somewhere comfy for the afternoon nap. For littleone doing a long (for them) walk in I would recomend having a ball for them to chase and kick up the path, they have fun playing with the ball and don't realise how far they have actually walked.
For the drive down, I like to stop at the service station on the A1 just to the north of the St Quentin turn off, I find the food is better than average (there is a Paul bakery for instance).
For eateries, I think that most are child friendly but it is weather the menu caters for what your littleone is used to eating, we find that La Taverne
http://www.latavernefontainebleau.com/ has a good menu. I don't know how adventureous your little one is but if they are more adventureous than the usual french childrens menu then we find shareing a main with an adult is a good idea. Just beware if you get a salad not as part of the set menus, they are massive.
None climbing things to do for littleones, There are good playgrounds in Boulancourt (near Buthier climbing area) and Nemour in the grounds of the Marie just opposite the Rocher Greau climbing area. There is also the Base de Loisir in Buthiers
http://www.base-de-buthiers.com/ which amongst other things has an open air swimming pool.
There aren't loads of things to do with littleones when it's raining though
http://www.tacot-des-lacs.com/ might be an option.
That was just off the top of my head, I might think of some things later...
Hope that helps.