Why would the fetus have any rights that are not automatically defeated by the rights of its mother if it's not a person? All anti-abortion arguments that I'm aware of start by arguing that the fetus deserves the rights of a person, and then proceed from there. Could you present a counterexample?

Sure, a couple I've come across:

"Potentiality" - this view argues that regardless of whether the fetus is a person, it has the potential to become a person, and thus inherits some moral consideration.

"Value of Human Life" - the argument that human life - at any stage of development - has intrinsic value and thus should be protected.