Legal ways to change the name of my daughter - she's carrying the name of her mother's former BF

I'm not sure about my case but I think it is very different with the others posted here. Couple of years ago my GF got pregnant & have a child but before my daughter was born we have a fall out & resulted to my child having a name in her birth certificate with that of his former BF.They are not living together & have separate lives & the mother of my child is willing to change her name to mine or her but her former BF wont give his consent for his own personal reasons. They are not married, he is not the father but because my daughter is carrying his name, he is somehow making hostage of the situation & not let go & not allowed that changes. Pls. any advise. It's the life of my child who is in jeopardy. Having a name of the person that have no relation at all with her but believe otherwise because of that name attach to her.
Appreciate it very much if someone who know the law could advise me what legal remedy i could possibly address.
Thank you.

good thing they didn't marry. the legal remedy i would choose would be to file a petition for correction of birth certificate, and use dna evidence, just like in the case we handled where we proved that the stated father was not actually the father.

Thank you for the advise attyLLL. Does it mean that even without his knowledge (meaning inform him), I or the mother of my child could file a petition for correction? If you don't mind but can you give me an idea how much the total expenses it would cost to do this correction? Thanks again.