Birth Certificate will be OK.

I had exactly the same problem with you.

We planned a trip within USA last December and I found out my daughter (6 years old)'s US passport was expired. There was no time to renew her US passport.

I called the security department of the airport, they said her birth certificate should be fine. The security did check her birth certificate when she checked in. So bring her birth certificate just in case and it will do it.