If anyone with a child and not living with the father is on benefits, there is no option but to tell the CSA who the father is, so the government can recoup some of the costs of raising that child.
Even when the situation changes and the mother comes off benefits, the CSA will expect the father to continue to contribute and will actively chase the money.
We don't know the circumstances and shouldn't judge. If he has refused to see the child in the four years since her birth and there has been no indication that he may, at some point in the future, change his mind - should the mother tell a toddler her father is x and she hasn't seen him because he doesn't want to meet her?
As I said in my first answer - the courts will not give any consideration to what the mother or the biological father wants. It will consider only the best interests of the child and that may include some form of contact. The court may also decide that the child is settled in a family life, is happy and will not benefit at all from contact at the moment.
To expect a four year old girl to go and stay with a couple of strangers is a nonsense and that will not happen. He is not going to get custody, unless you as the mother are seriously neglecting the child, or putting her in danger and there is no other family member to care for her.
Your partner could consider adopting your daughter. You do not need to be married. As her biological father has no parental responsibility, you were never married to him and his name is not on the birth certificate - you don't not need his consent. However, this is not something to be done lightly and will need a lot of consideration by all of you.
http://www.adoption.org.uk/information/step_ad option.html