It's highly unlikely. If the father is truly out of the picture the person you are choosing to have custody of your daughter may be able to adopt or petition for temporary custody by a close relative. the person who you choose must be within 3 degrees of consanguinity of your daughter in order to make the process as easy as possible. But even then, the burden on them to effectively terminate the father's rights is a high one and difficult to satisfy. Whether or not a court would terminate a parents rights is very fact specific question that an Attorney can help answer for you.
Unlikely is what I would say. The best interest of your daughter should control and if the person you named fights in accordance with what you indicate in your document, that person might be able to carry the day in court. That person will definitely need the assistance of an attorney. I would not suggest doing this on her/his own. If your new husband wants to adopt your daughterm, you may be able to proceed with that and avoid all that potential problem.