With the child still in contact with the birth father, and him being current on child support, you will have to prove that he is an unfit father to have his rights terminated. That may be difficult if he has other children that he is raising.
As far as the name change, you will have to be married before that COULD occur, and the biological father does have a right to contest it, as well as block it if the child currently holds his last name. Since he has been proven to be the child's father you will not be able to legally put your name on the birth certificate unless you do a stepparent adoption, after you are married. Getting the father to give up his rights will be easier than fighting him in court and proving him unfit.
Bottom line the father has a right to fight anything you wish to proceed with regarding the child, unless his rights are terminated, either voluntary, or by the courts.
The temptation of saving nearly two thousand dollars a month may be the only thing that you have to offer the father at this time to get him to agree to an adoption, IF that is what is truly in the best interest of the child.
If this is more of an issue of the parents and adults not liking one and other than counseling for the family on how to deal with a stepparent situation may be more in order for you than a stepparent adoption. Forcing a parent out of a child's life usually ends up causing more harm than good. If the child is not being abused or neglected in the home of the father, there is little that you can do to stop contact.
My advice would be to begin to document everything. How often there is contact, both physical and over the phone, how the child reacts to the contact, how the child's school performance is after and directly prior to contact and so forth. This can all come in handy in case your situation needs to go to court for whatever reason, adoption, or a custody battle.
Stepparent Adoption Blog