If your name is on the BC I would have to presume you and the mother signed an acknowledgment of paternity. Unless somebody wishes to challenge that, I don't understand your suit to establish paternity. The AOP is a legal acknowledgment that the people listed are the parents.
So, why the paternity suit? The courts are obviously treating you as the legal father evidenced by the fact you were granted any rights in the custody/visitation action.
If you want to change the name, you will have to file a petition in the courts asking the court to agree. The mother will have her chance to agree, object, or merely comment on the issue. The court will determine if a name change is appropriate and what the name will be. If there is disagreement, it appears many courts seem to prefer a hyphenated version of the two parents names.

