@LittleMySister
You are quite right in that the child deserves to have a proper relationship with both parents. I would wonder how the child develops a perception of what it means to have a father when she had never experienced living together with that person in a family unit and has spent very little actual time with them.
When this child is told that this is your father, what do those words actually translate to her as an actual experience? Does he do all or most of the things with her that her mother does as a parent?
Probably not I would think . So then what does the term "father" mean for this child or any child in a similar situation. She is a bit to young to understand the biology of conception, and the dad has not spent enough substantive time with her to fit within her definition of a parent in the same sense that her mother is her parent.
Someone saying to you " this is your father'" of a man with whom you have no real sense of a familial connection and not a particularly strong bond is not telling you anything that in and if itself would inspire an immediate or strong sense of security or comfort.
A child can enjoy spending a fun filled Fay with strangers, and she can even seem to like them or have a positive reaction to them. However, that does not mean that she wants to stay overnight in their home or spend any considerable amount of time with them without her parent being present.
The OP's partner may think that he has the feelings of a father but until this child experiences him as her father in any meaningful way there will never be a parent/child relationship developed.
Sadly, biology made him a father, but it is only considerable time, effort, and hard work that will make him a parent.
The OP may be a wonderful person, but the child seemingly has a perfectly adequate mother/parent, and what she needs now is a perfectly adequate father/parent. That is not a we or us job ,that is a position that the father alone must attain before he has the stature to bring his child into his we/us relationship with the OP.
The OP may be well intentioned, but she, at this point , is trying to insert herself into a play before her character has a role or part to play.
Simply being present on the stage does not mean that there is any substantive role for you to play in this act.
OP should step back and let the father work proactively to improve and develop his relationship with his daughter if he wishes to become a father in any meaningful way.