You also point out that the mother is to blame for this bad situation. That may be true, but blaming doesn't help that little girl. It is too late to change what happened 6 years ago. The question at this point will become this: What is in the best interest of the child?
Chances are, whether you spend ten thousand dollars or not, a family court would conclude that the child should continue to reside with her mother but that she should have parenting time with her father. I say this not because I think it is fair or that I think it is best, but because that is very typical of most family court decisions in cases like this.
Is it possible that the mother is allowed by the court to continue excluding the father completely? Yes, but the odds are against her being able to do that unless she has some reason such as a criminal history. Is it possible that the father can convince the court to give him custody due to the mother's history of poor co-parenting (to say the least)? Yes, but don't count on it. Getting a good lawyer increases your fiance's odds in either case.
Is there an existing court order for custody? If not, the father legally has as much right to the child as the mother does. He technically has the right to pick the child up from school and refuse to communicate with the mother. Not only would that be traumatic for the child, but it would certainly backfire because the father would lose the only high ground he has once the court does get involved.
Talk this out together in terms of what is best for the child in your opinion. It sounds as if you already believe that it is not best to let the mother do what she wants by cutting the father out of the child's life. I agree with you on that, but what IS right? Should the mother have primary custody while the father has regular parenting time? Should there be equal 50/50 parenting time? (That is my suggestion.) Or should the father have primary custody?
Once you have decided this, you need to get a lawyer and file a petition for it in court. Don't worry about the odds of success or the cost. Pay what you can for a lawyer but don't let anything slip away because you can't afford it. That is foolish. The only reason to have a lawyer is to enhance your chances. If you don't ask for something, you have almost no chance of getting it. If the lawyer tries to discourage you or if you can't afford a lawyer, file it yourself.
Technically the fact that he is a male is not a valid point. Technically the fact that they weren't married is not a valid point. Some judges realize this and give you a fair shake. Others do not, but there is no excuse not to try. If you don't try, don't blame the judge, the system or the lack of money.