I would definitely contact a lawyer. I don't believe you will be able to terminate his rights, or get full custody without notifying him of your attempt to do so. Even if you get full custody, this would not necessarily mean that he wouldn't be able to have visitation at some point if that is what he wants. You would need to petition the court to terminate his parental rights. If he decides that he wants to assert his parental rights, you will likely not be able to stop that unless you can prove that she would be in imminent danger while in his care. If he truly has no interest in parenting this child and he understands that if he fights your petition to terminate it will meant that he will at the very least be required by law to pay not only child support going forward, but could also be on the hook for child support owed for the last 15 months, he may decide to sign away his rights. Either way, I think it is in all of your best interest to get a legal and binding custody arrangement.
That said, you should think long and hard about what your motives are for wanting to cut your daughter's father out of her life completely. Even if things are going well and you are perfectly capable of taking care of your daughter without any help financially, it helps to have a little extra both financially as well as help physically. Bottom line is, you didn't create that little life alone and the other person has a duty and a responsibility to provide at the very least financially to the care of that child. However, t would be so beneficial for your child to have a relationship with her father if at all possible, if it is a positive and safe situation.