We got charged once at a buffet when my older DD was 2 (can't remember if it was Crystal Palace or Chef Mickey's). She had just turned two a couple of months before, so it wasn't even close, but she was/is tall. I carried a copy of her birth certificate with us (not an official copy - just one copied on the printer at home). I told the server she was still two and offered the birth certificate, but he just took the charge off without asking to see it.
We also got asked her age at one of the turnstiles, but I said she was two and again offered the copy of the BC. The CM didn't ask to see it - he just waved her through since she didn't need a ticket.
When we went a few weeks before our older DD turned 10, I brought a copy again since she looked about fourteen and was almost as tall as I am.
You could make a copy of the passport front page if it makes you feel better. I expect the CMs will take your word for it, though.