I am confused why your friend thought good social justice arguments do not use logic to defend their claims. Good arguments of any kind use logic to defend their claims. Ergo, all the good social justice arguments are using logic to defend their claims. Why did you not say this to your friend?

EDIT: Also confused about your focus on axioms. Axioms, though essential, are the least interesting part of any logical argument. If you do not accept the same axioms as your debate partner, the argument is over. Axioms are by definition not mathematically demonstrable. In your post, you stated that axioms could be derived from other fundamental axioms, which is incorrect. Could you clarify your thinking on this?