While I agree with pretty much everything you’ve said my K2’s are here and we didn’t have any documents from their mother giving permission. Not even a hand written letter. My husband has always had full physical custody of the children but nothing ever legalized. They weren’t given a DNA test or anything. They just asked if their mother is aware they were moving and that was it. We are going to get something official just in case it comes up later though. I think it’s just at the discretion off the officer.