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I'm so happy to see that you didn't need it. Yay!!!
In this situation if they want to pass on US citizenship to the child after entry they will need something that states the father has custody of the child too.
It does not apply to you. It would be if you had a previous marriage and child from that marriage. Then they want to see that you are paying child support.
Use the marriage certificate as the proof needed to make the update for SSN. It does not matter that the name on it is her maiden name or that she is using a combo of names.
It's because she assumed her married name. It is not an official name change that needs a judge to sign off on. That is why she just has the certificate in the married name. Even if it is a combo of the two names.