As to the legal issues involved, your husband should consult a lawyer. As for the moral issues, he is responsible for this child and not just for her financial support. If the legal circumstances and the child's own best interest indicate that it is not possible for him to visit the child, then that is one thing. If he should refuse to consider his own daughter to be his family, then that is quite another. If he does so because you have made it clear that you do not want the child considered to be family, then you may well share moral culpability for that act of depriving an innocent child of her father.
The fact that your husband had an affair no doubt continues to cause you pain, especially since the affair resulted in pregnancy. But you cannot salve your own pain by inflicting either pain or injustice on this child. She is your husband's daughter. She is your own stepdaughter. Even if the two of you cannot be a part of her day-to-day life, you both should consider her family, to such an extent that you would willingly and unhesitatingly bring her into your home should anything happen to her mother and there be no other provision for her welfare (such as a legal guardian named in her mother's will).
Marriage is a sacrament that is supposed to open the couple to new life, even when that new life is not children born of their own conjugal relations. The unselfishness of that openness not only benefits any children dependent upon the married couple, but also benefits the married couple and their own relationship. If you allow selfishness to deprive this child of her father, that very selfishness could also poison the marital relationship that you and your husband have worked so hard to repair.
I strongly urge the two of you to get marital counseling, even if you have already had counseling in the past. The Pastoral Solutions Institute
, a Catholic counseling apostolate, may be able to assist you further. God bless.