Well social/religious conservatives often think the child has rights even during pregnancy so it's not as simple as the mothers rights.
The libertarian view tends to much more favour the parents rights to make choices for their children if I remember correctly, and obviously favour the option where the government isn't deciding for them.
Right, is there a non-religious reason to be against the right to choose to abort early during pregnancy?
Exactly.
My personal belief is that life begins at conception. As a result, I’m opposed to abortion in all cases.
… but I’m also an anarchist, and therefore believe it is emphatically not the state’s role to make these types of decisions for people.
I don’t think there is a “right answer” here in terms of policy. Some large portion of the people will see it as a violation of their rights no matter how extreme or nuanced the line is drawn.