Interesting open and many good and interesting responses.
From where I sit I figure stay with the child at least. Even at the start you refer to the child as "his 2 year old daughter" that says it all. It's his daughter in every way that really matters. Okay the relationship has a problem although how much is unknown from us. They'll have to decide that. To pack up and leave the child would be hurtful to the child as she knows no other father.
Now my own experiences. I have 2 daughters and a son. The 1 daughter is biologically mine no questions asked no doubts. The other daughter is a mystery she was created by either me or my friend. at the time my then wife and me were in a 3 way with a friend of mine, she became pregnant and that man and me both had a hand in raising
our little girl. (my ex eventually married the other man). I fell in love with that child the moment she was born and my friend did as well (it was a crowded delivery room).and I don't care about the DNA.
My other child is my son who I just recently was introduced to. His mother and me went in different directions before she even knew she was pregnant (a very short lasting marriage). She met and married another man and he helped raise the child as theirs. I met my son recently when he decided he wanted to find and meet me, he was with his parents and when we've talked since, he refers to that man as Dad, he refers to me by my first name and although I'm a biological link, I'm not really his dad, I didn't raise him.
Okay I didn't have a spouse or partner lie about who the DNA father was (even my ex let the man who raised my son know the truth before they were married) but I can tell you that he should at least stay in the child's life if at all possible and what's your nonsense about "forced to pay child support"?
He was perfectly willing to pay her expenses before, clothes, food, a home and he's been tagged on the birth certificate. As others have said there's more to being a parent than DNA ask the millions of adopted people that have been raised by men and women that they don't share any biology with.