To piggy-back on what Malie said (hopefully someone will clarify or confirm) if I'm not mistaken, you need to make it clear to the SS agent that you are applying for a SSN for a permanent resident
, as opposed to a citizen. Unless you make this perfectly (and repeatedly
) clear to them, they will likely assume that you are talking about a SSN for a citizen, and at this point you unfortunately are unable to do that because you don't have federally-issued documents to prove citizenship (passport or COC). You do have documents to prove Permanent Resident status. When you get a SSN for a permanent resident, it will be issued in your child's Korean name, as Malie said, and after you have federally-issued documents to prove citizenship, you go back and have it changed.
get a US passport without a SSN, though for some of us, that has caused headaches, too.