Just my opinion and I have no kids nor do I have any experience with this! But I guess I am a little more traditional and I think a child should have their father's last name, especially if the child's parents were married. A friend of mine always used her maiden name for work but her daughter has her husband's name - which is her biological father by the way. Another friend divorced when her son was 3 and the child kept his last name while she went back to her maiden name. And another case when the parents didn't marry, the child has the father listed on his birth certificate but the child has his mom's maiden name as his last name. She has since married and took her husband's name but her son's name was not changed.
I agree with asking her what she thinks - she may have a strong opinion about it and I think that's valid. If I were you SG, I would not keep the ex's last name.