I have never seen a CM refuse entry to a child. I have heard a couple of them sweetly ask "How old are you?" (and part of me wonders if they were trying to get the child to say, "Mommy told me to say I'm 2 but I'm really 3!") I think you'll be fine especially since your child *is* two. You're not trying to break any rules.
On other hand, if I were in the situation, I would make a photocopy of his birth certificate and just tuck it into your wallet. It's easy to do and would give you some peace of mind. I seriously doubt you'd be asked to show it, but I know (if I were in your situation) I would worry about it every time I went through the gates, and that would suck the fun out of the vacation. It's easier just to carry the proof!