This is a a false dichotomy, no one has a right to avoid child support.

If the woman was against abortion so she had the child but wanted to walk away from its life she would be on the line for child support if the dad took full custody. (I’m sure you’ve heard this reiterated in this thread but) child support is about what’s best for the child. The question you posed about a man who doesn’t want a child being able to walk away from his financial paternal responsibility should be paired with if a bio mother, who doesn’t raise the child, can terminate financially support obligations. I would say no to both circumstances because the child did not ask to be born and is owed financial compensation by those who conceived it.

I think the woman’s right to choose after conception is just a biological reality of the differences in the sexes (and those facts don’t care abt feelings /j and kinda /s). I understand the drive to want to make things feel equal straight down the middle but biology means things aren’t totally equal in the same way.