K-2 visa problem: In the birth certificate of my Ukrainian fiancee's son, the name of the father is a fake name.

When applying for K-2 visa, I understand that my fiancee will need her son's biological father to sign a statement that gives up his parental right and let the child migrate to the US. When the child was born, the father didn't want to take any responsibility. Taking advice from others, my fiancee made up a fake name for the baby's father on the birth certificate. Now she can still ask the biological father to sign a statement to give up his parental right but his name is not going to match what's on the birth certificate. I am sure this will raise a red flag during the interview with the US Embassy in Kiev. What should we do? I'd greatly appreciate advice from any legal professionals whose clients have had similar experience.