My GF lost her Korean passport and she needs to get her family registry, etc.

You said her previous marriage to a US citizen was abusive. VAWA might be a possibility; though that depends on whether there is evidence of actual abuse. (Of course, if you are a US citizen, then applying for a green card based on marriage to you is simpler, but the problem with that is the lack of ID for marriage.)
She’d still need ID for a VAWA application, wouldn’t she?
 
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