Are there legal consequences of giving our baby the father's last name?

There should be no particular implications regarding custody or placement if you give your child the father's family name. And it's likely a good idea. You are entitled to seek child supportwhich after all can be a big help with the costs of raising a child. If the father wants to demand some degree of placement, he can do so whatever name the child bears.

Custody is not at issue. I assume you and the father are not married. Will he be on the birth certificate by signing an affidavit of paternity. Will he file with the court to establish himself as the legal father? Will you? Will you (or the state if you receive any public benefits) file for child support? In other words, what last name you give the child has very little significance and doesn't change either of your rights or responsibilities. What you do re establishing his legal paternity is what counts. This should be done as quickly as possible after the child is born so as to establish whatever relationship there will be between father and child and to provide support for the child. Of course if you don't want him in the child's life and are not dependent on anyone else for support, then don't call him when the baby is born, don't tell him your whereabouts and don't give the child his name. All of that will leave it up to him to go to court and establish his rights and responsibilities, which he may or may not do.

The name will be evidence of paternity. His departure will have far greater consequences for determination of custody.