Here's the deal. I'll have to assume that bio dad never formally relinquished his parental rights, he just agreed to let you have sole custody. So as it stands now, should you die, the child would automatically go to the bio dad. If you are just wanting to change her name, you must file a petition with the court and bio dad must be notified and has a right to put a stop to it. Should the court grant the name change and you divorce your husband, are you wanting to do this all over again? And just because the name would be the same, it does not change the fact that your husband would never be the child's legal father and the child would never receive any inheritance unless he adopts.
Should your husband decide he wants to adopt, then he would need to petition the court for a step-parent adoption and the bio dad would have to be served and notified. He can also put a stop to any adoption. Should by some long-shot the court terminate his parental rights and grant the adoption, then the name change is provided free of charge in the adoption decree and the child is now the legal child of your husband's. That includes should something happen to you, your husband would have sole custody and the child would be entitled to any inheritance from him.