Hello Breakfast welcome here!!
Although the father is not supporting the child ,he still got his name on his birth certificate,so even the custody is awarded to your fiance there would be a problem with the visa as far as sole responsibility is concern,...my suggestion is for your fiance to gather all the documents to prove she's the only responsible with the childs day to day basis like school certificates,parish letter on where they attend service,letter from GP's (pediatrician) ...and if the father of the child is really not concern with the child anymore as far as financial support is concern, a letter from the father(paternal consent) stating that he has no objection if the child settles ''for good'' under the care of his mother.