Well no, axioms are not typically a thing in philosophy. In philosophy we are concerned with trying to figure out what's true, not just declaring random things are "axioms" and thus true unless proved false. The latter approach would make just about anything an equally "valid" candidate for truth, and suggest, contrary to basic principles of reason, that we should believe things without having sufficient reasons to believe them.

With that in mind, the question has to be: why should we take your antinatalist principle to be true? Or perhaps, what makes it more plausibly true than the other things people typically believe that it conflicts with, e.g., that life has inherent value, that procreation is an inherent right, or that happiness as well as suffering is morally significant?