Having never flown WN with an infant, I have no personal experience, but I find it very hard to believe they would not accept a valid passport. It is every bit as authoritative (and, as the OP points out, far harder to fake) than a birth certificate. I have never had any agency foreign or domestic refuse our children's passports as a valid form of ID and proof of age.

Indeed, my own personal birth certificate is from a third-world nation and looks like something a 6-year-old drew. I have had several agencies question its authenticity so make a habit of providing alternative documentation whenever possible.