Hi,
There are a TON of things at play here.....You don't say how long you have been dating, just that you were friends for three years and have a five month old. Do you live together? How long have you each been divorced? Does he have kids from his first marriage? Have you talked about what went wrong in each of your marriages and what part YOU played, retrospectively, in the demise of the relationship?
These things are important to know about each other. so you know if you have dealt with the issues within yourself AND to know if you have dealt with the issues from the other person. WHY doesn't he want to get married? Because he doesn't want to have things end terribly and destroy lives? Or because he thinks he may want to be with other women in the future and can't imagine spending the rest of his life with just you?
These are very different issues and the mean different things.
When you have these discussions about what YOU want, how much of it is focused on what HE wants? when he "shuts down or freaks out" what do you do? do you tell him you understand that he has concerns and you would love to hear him out so you can reassure him?
You say if you didn't have a child you would be willing to wait longer to get married, but you don't want her to grow up and you not be married. But she ALREADY has 2 parents who are not married. That is a choice that has already been a made.
As far as the future.... being married is something that you DO NOT DO FOR KIDS. You do it because you and another human being want to build a life together. You want to take care of each other. You want to sacrifice to make the other happy. You want to share joys and sorrows and happiness and sadness. You want to love when you feel you can't and when you know you can. If you only get married so you daughter has 2 parents that are married you are not entering into marriage for the right reasons.
Most importantly, if one person can't (for whatever reason) enter into marriage for the right reasons it is a disservice to all involved to try and get them to do it.
I would back off. Build a relationship with him and raise your daughter together. Learn about him. Find out what his issues are. Does he want to deal with them? You can't make him. He has to want to.
You can have dealt with issues from your past, but know that you don't want to have issues again. He has to feel like he can be a good husband and that you will be a good wife before he will commit to getting married.
Perhaps some counseling for you will help you come to terms with having an expectation that may not be met and what you will do if you are not willing to change to be with the father of your child in any way other than the way you want.
Good Luck.