How would you convince your ex that your daughter deserves to have her dad's last name as well?

I agree...a child should have their fathers last name. I tend to be old fashioned and I also think that there are far too many women in situations where they aren't with the father, that think they OWN the child/ren. There is obviously no reasoning with this woman and for your childs sake you need to keep your relationship civil. Before your child get old enough that all this will cause trama to them, take this to court and see what you can do. The mother has no right to keep you out of the decisions about this child...your child is not a posession. If you win, great, if not, leave it alone and just raise your child the best you can and know in the end, it doesn't really matter, what matters is being a good dad for your child.