I am from Michigan originally (exact same area).
Paternity laws are weird in Michigan. You need to consult a lawyer familiar with paternity (not divorce, treated differently and I don't know any, the one I was using retired). You will need solid evidence for violence, drug abuse things like that... Female custody rates in Michigan are very high. There are some laws that are anti-unmarried father. In some ways, you are better off if you were married with Mom. That offers you more legal protections in Michigan. Not to scare you, but you need to know what you are up against. So you need a good paternity lawyer.
You need a DNA test. Birth certificate is not good enough. If you do not have this filed with court, you are currently a legal stranger to the child. I would settle that with Mom before doing the whole "break up" thing. I mean, you just want to be the child's father officially. It will put you in a better position later and in general. I would say that you need a strategy. Make your moves very carefully.
Also, and this is just general advice, have you tried counseling? Other relationship fixers? If you make relationship work, you are certain to be with her every day and influence her upbringing. Custody fights are really exhausting and destructive. Of course, I do not know your situation.
PS-I think time limit applies to married men, who have raised child as their own.