Plus someone could have revealed his name in the births section of a newspaper

I don't really think that's a goof. He was at the hospital and we don't see every single scene there. It is possible that he has seen baby Mikey when he was with the other babies. And usually the baby's name (and mother & doctor's name) are written on a little sign on the bed, so maybe James went to see the baby after Mollie got a room. So he might know the name from that.
"Life is like a movie. Only you can't pick your genre."