Child support has nothing to do with a child's last name. Her name can be changed to She-Ra, and child support would remain the same.
As far as obtaining a name change, it's easiest to do if Dad consents. Elsewise, you'll have to prove that it's in the child's best interest to have her name changed. I expect that it would be harder for Dad to contest, since she bears your name and not his.
The courts rarely address young children in matters of name changes. The circumstances would have to be pretty extraordinary, and nothing you've said here would fall under that heading.Would a judge speak to her or is she too young to?

