6 is FAR too young [in my opinion] to choose what name she wants, if she wants to change, I would wait until she is older and more grown up. Six is a baby.
Let me get this striaght...does she not know that your DP is not her bio father?? Do you think maybe when she finds out this, and that you changed her name to his that she may resnt you a little for it? Kids very easily get sentimental for what they don't know, and for me, this would terrify me. If she's not old enough to understand about her father, she's not old enough to make a decision on her name. I think she needs to know all the facts before you do anything.
Secondly, especially as you are not married, would not change her name, or give the new baby his name. Our children will take both of our names and we are married. If I was unmarried, my baby would take my name. I would not give the most precious thing in the world to someone else, if he had not taken the step to give himself to me. I'm sorry if that sounds really harsh, but it is my opinion.
There is no rush on name, my friend has a DD [age 10] with a different name to her husband and baby, and dd desparately wants to have th same name as the rest of the family, but it is not so easy with the father...especially in your case if he is absent.