Take one of the axioms for your system of morality (I don't mean a consequence, but one of the core values).
Then tell me one thing, that if it happens, would prove your core value is wrong.
I believe I already did this many pages ago for both Kantian-ish ethics and Utilitarianism. You were unsatisfied then, and I wouldn't expect you to be satisfied now. So why bother?
But here's another idea: for purposes of this discussion, my morality need not have any
axioms or core values to begin with. If this strikes you as odd, think of doctors as an analogy. They, too, need not define key terms like "health" and "illness" in an axiomatic manner. When you get down to it, medical science runs on intuition, seat-of-the-pants assumptions, ad-hoc models for special cases, and similar philosophically-mushy concepts. I don't think there are any generally-accepted axioms of medicine. Nevertheless, medicine is an objective business, with but minimal niches for what you might call "medical relativism". (Aspergers disease: is it really a disease or just regular nerds being pathologized arbitrarily?) I'm perfectly willing to run with the idea that moral philosophy is like that. But it would still
be an objective
business. Health and illness would still exist independent and external to the mind, and would still be based on observable phenomena.