Also sharing my experience.
Child was born in Europe, got him a passport + english birth certificate and two "2 x 2" inch baby photos for the green card.
Contacted Airline (SAS) regarding boarding a baby without VISA/ESTA, was advised to bring official documents that clearly stated my child was visa/esta exempt, and arrive early for our flight.
Airport
Arriving 3.30h before departure, a helpful lady at check-in asked if my child had any VISA/ESTA. I explained our child had intentions to stay in the US for more then 90 days, and thus disqualified for ESTA. I presented our travel documents, ( green cards, passports, birth certificate, CBP's official "carrier information guide", the FAQ page from the American embassy in Japan regarding no visa requirements to LPR child born abroad ) the lady took pictures of all our documents and sent to her supervisor. She asked if we had any official document indicating this exception with our child's name printed on it, I declined and said the papers we had was official but not personal, only general in nature ( CBP carrier guide, Embassy FAQ ). The lady told me they had to make a phone call and it could take a few minutes, ( I assume they called the Regional Carrier Liaison Groups (RCLG) whom assist mainly overseas carriers determine U.S. entry qualifications. ) and after 30 minutes, we where cleared to board.
Arrival at LAX
CBP officer was somewhat confused when we presented our baby's birth certificate + 1 printed page from CBP's own website indicating that baby born to LPR's parents temporary stay abroad does not require VISA. He went and talked to a supervisor, then came back and asked if our baby had ESTA, to which I again politely replied our baby does not qualify for ESTA based on intentions to stay longer then 90 days ( I assume even the most argumentative CBP officer will accept this answer, as did ours ). The officer made another phone call and then stamped our babies passport as admitted and wrote NA3 under the stamp, and then we followed him into a separate processing area.
Was eventually called up to a booth, where a new CBP officer immediately asked for a birth certificate and pictures of our child + our green cards and passports. He gave me a piece of paper and told me to write our address on it, eventually he stamped our baby's passport with the temporary I-551, and said a green card should arrive in a month or two. He handed back the original birth certificate, I hesitated and told him, "I was under the assumption USCIS need the BC original?" , he declined and said not anymore, they used to lose them all the time, so copies are sufficient ( not comforting ).
I asked to receive an Alien number for my baby, so I could follow up with USCIS in those rare circumstances our child's green card "won't make it", the officer declined my request stating that they don't issue A-numbers and pointed to the I-181 where the field was left blank, and stated USCIS will fill it in. I asked how would I be able to contact USCIS without an alien number?, his eye's looked a little blank, and I did not feel motivated to argue with an officer who had filled in several of these I-181s on previous occasions.
Looking back I should have asked for a copy of the I-181.
Planning to see if I can get my child a SSN in a few days ( I believe their handbook states I-551 is sufficient, but in reality? - who knows ).
