...I have an average sized almost 2 year old, and I carried her birth certificate when we flew in case I was asked to prove her age as a lap child.
What is the big deal about carrying a birth certificate? You can easily get copies of it if you don't want to carry the original. In WA, they are $7. I wouldn't dream of traveling without ID for everyone, even if it isn't a photo ID.

In fact, you could even make a photocopy of the original since you really don't need the seal on it just to show proof of age.