Can I request her original birth certificate with my last name and no father listed and just cut him out of her life for good without legal effects?

My daughter's father was not on the original birth certificate as he was not at the hospital when she was born. He requested to have his name on her birth certificate. We filed the paperwork and paid the fee and the only thing missing is a copy of his ID. He has not been responsive in providing the information needed to have the name change. I found out that it could be because he secretly married a woman who lives in another state in April for his green card and was fully intending on living with us in here without disclosing this information. My daughter gets Medicaid and I receive food stamps. He does put $80 - $100 a week in an account for her.