I hear that you are not trying to be selfish and lessen the stress for yourselves, at the expense of the child. You think it might be better for the child to never see your DH, than to suffer the endless conflict that results from him attempting to see her.
Nevertheless, I think your DH waiving visitation would be a horrible mistake.
This child has a mother who makes abominable choices that put her (the child) in danger (i.e., living with a man who has already molested a pre-pubescent girl). Assuming there is nothing wrong with your DH that would make a reasonable mother try to keep him away from his child, then this mother is also someone prone to extreme conflict without reasonable provocation; vengefulness; and prioritizing her own wants over her daughter's needs (i.e., her hatred toward your DH is more important than the child's right to have a relationship with her father). You guys cannot, in good conscience, walk away from an endangered child - especially when she is your husband's own!
This child will be exposed to conflict, stress and misery from this mother, whether your DH leaves the picture or not. If he leaves, the mother will create conflict with someone new, or keep the current conflict alive, by telling the child about the no-good, deadbeat Dad who never loved her and walked out on her.
You may think, "She's telling the child these things about my DH, anyway!" The important difference is, even if the child only gets 4 hours a month with your DH, as she gets older, she will be increasingly aware of his presence, his love and concern for her, and that he's fighting for more access to her. When she hears ugliness from her mother, some part of her mind will say, "But that's not true. When I'm with Dad, he's not like she says." But if your husband waits to tell her he loves her, when she's an adult, it will not have the same effect. She will have spent all her vulnerable, formative childhood years BELIEVING what her mother tells her about him, BECAUSE HE ISN'T THERE TO TELL HER OTHERWISE. And she will wonder, "If you loved me, why didn't you fight for me? How could you give up on me? How could you leave me with my crazy mother and not try to protect me?"
Whether you have a good relationship with your own father, or whether you just longed for a good father when you were young, wouldn't you agree that children hope to be cherished and important enough to their fathers, that their fathers could NEVER stand to walk away from them, no matter how hard things were? Your husband needs to be such a father, to this child.
Also, if he keeps fighting to enforce his parental rights, in time two things will happen:
1- He will be awarded more visitation. Unless a judge is going to take a baby away from the mother altogether, it is hard to award visitation that forces a baby/toddler to spend a lot of time away from "Mommy". The same is not true, with school-aged children. Many states are adopting standards in which older kids spend fully half their time with the "non-custodial" parent. Where this is not the case, every other weekend and at least one day during the week - EVERY week - is pretty standard.
2- The court will take more and more exception to her refusing your DH access to the child and/or ignoring court orders, if she keeps doing that. My husband's ex was, in many ways, like yours (but without the child-molesting boyfriend!) He fought for access to his son, and to enforce the custodial orders the court gave, but which his ex wouldn't follow. This began when his son was 6 months old. It was miserable. After eight years of it, the court awarded my husband sole custody - because of all the mother's efforts to deny him access to the child and to interfere with the father-son relationship. I would think your DH would have an even stronger argument to get custody at some point, if the mother continues endangering the child by living with a child molester.