I don't know about AA (we've been mostly on UA and SW) but IIRC we've definitely been asked for his proof of age (at check-in, not security), maybe 50% (?) of the time. Given the age of your lap infant (close to the cut-off date) I would definitely carry something, even a copy, rather than get grief about whether he/she needs a ticket!

Our kids now have passports, but before they did we carried our extra original of their birth certificates - I also recommend ordering that second copy as a back-up. Or you can get them passports too - but you will need to mail in their birth certificate (they send it back to you, but again, good to have a backup....) as part of the application process.