I went through something similar, different route though. Take all evidence with you to the airport and go in early. The check-in agent will most likely have no clue about this situation and will need further clarification.
In my case, a US homeland security agent had to come to the check-in counter to complete the check-in. The airline said while we understand that the baby technically can get visa on arrival, but we can't check her in without a Visa number or Green card number or ESTA.
Otherwise, It wasn't too bad, but we did go to secondary inspection in the US and waited 2+ hours for our daughter to clear. Take copies of birth certificates and passport pictures of the baby, they'll ask for them in the US Point of entry