What makes it a good question? It's obvious that we can't conjure universally compelling moral axioms out of nowhere; all of our moral reasoning ultimately rests on fundamental values that one either cares about or doesn't. If our sceptic has the capacity for empathy with animals, we can try to convince them to elevate their empathetic response to the status of a moral value. But if they simply don't care, and insist that they will only begin to care if logically compelled to do so, we're in the same position as if we were trying to reason a sociopath into valuing other people's wellbeing as an intrinsic good.
I think, first of all, you're not wrong, but also, you're mostly wrong. People with completely separate value systems exist, but only theoretically, or historically. I know there's a lot of hand-wringing at the moment about society being fractured and divided, and it may be Pollyannaish of me to say this, but I think we're mostly all on the same side with approximately the same bunch of deep values, by the power of culture (and we more or less all come from the same culture, "modern"). We can have productive discussions, we can work it out. The Myth of the Framework is a relevant book.
