Quote Quoting Mr. Knowitall
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It's easy enough to order a copy of your child's birth certificate.

It's not clear what the issue is. You appear to be stating that you successfully amended the child's birth certificate to reflect his name change. If dad is subject to a child support order for your child, that order continues to apply even if the child's name has been changed. If dad is in prison, dad's not in a position to bring a petition to modify custody.
I agree, and if dad destroyed the child's Birth Certificate then just go down to the office of vital records and get another copy. They probably will charge you something for it, but they are easy to replace.