In general, a court is much more likely to order you primary custody, with limited visitation rights to the other parent.
In order for you to get sole custody, you would have to demonstrate to the court that the father poses an imminent danger to the infant, and that it is not in the best interests of the child to see the parent.
There is a procedure for names changes in family court. In addition, if you remarry, you could seek to have your daughter adopted by the new husband, which would terminate the father's parental rights.
You haven't stated whether you were ever married or whether there is any judgment or order already in place regarding custody, support, visitation or paternity. These factors can make a difference in whether you actually need to do anything or just wait for the father to try to claim his rights at some time in the future.
It sounds like you might want to sit face to face with another attorney if these things have not already been discussed. Good luck.
Child support Child custody Considerations in child custody decisions Shared custody Family court and child custody cases Best interests of the child and custody Physical custody Sole custody Child support and custody Paternity and child custody Visitation rights in child custody agreements Parental rights in child custody Family law Paternity
Sign up to receive a 5-part series of useful information and advice about child custody law.