A unicorn is defined according to its physical characteristics.
But unicorns can only be seen by "true believers". No True Scotsman. Unicorns are undetectable under normal circumstances. The fact that you have never seen a unicorn only proves you are not a "true believer", it does not disprove the existence of unicorns.
Morality is not defined by physical characteristics. Morality is abstract. Empiricism works on tangible things, like unicorns, but not on abstract stuff, like morality.
Abstract concepts like "height" and "weight" and "mass" and "speed" all function perfectly well as EMPIRICALLY VERIFIABLE PHENOMENA.
Some abstract concepts are Quanta and some are Qualia.
Quanta are quantifiable, rigorously defined, scientifically verifiable phenomena (and emotionally meaningless).
Qualia are qualitative, experiential, broadly defined, personal opinions (and emotionally meaningful).
We don't need to prove moral realism.
By definition, things that are considered "real" are independently verifiable. Anything that is not independently verifiable is imaginary (conceptual).
All we need to do is determine whether it's more rational than moral non-realism since one of the two must be true by law of excluded middle.
Simply by presenting mutually exclusive options does not necessarily mean one of them is "true".
For example, (IFF) Vishnu is a god (THEN) Nanabozho is not a god. (IFF) Nanabozho is a god (THEN) Vishnu is not a god. Mutual exclusivity does not demand that one of these options is "true" and there is no way to determine if one option is "more rational" than the other.
Having a moral principle be "axiomatically true" better evidences moral realism than non-realism.
This is demonstrably false. Axioms prove nothing. The statement, "Gurglantuabatory = eating a purple dinosaur + punching a sparrow + while riding a unicorn" is axiomatically true.
Moral realism is both intuitive and rational. They needn't be mutually exclusive.
`Claiming something is intuitive does not demand evidence. Intuition is personal and Qualitative.
Virtually anything can be considered "intuitive". A native from the jungles of Peru would say it is "intuitive" to track an animal who may have passed through several days earlier. They might not be able to explain exactly how they know how to track the animal, but since they do it all the time, and learned it as a small child, it is second nature to them and properly "intuitive". Claiming something is "intuitive" is exactly the same as saying "I know how to do it or what it is, but I am unable to explain it to you". It is a naked appeal to ignorance. I know Santa Claus is real intuitively and if you doubt me it is because you are a bad/mean/unimaginative/insincere/stupid/irrational/unreasonable person.
`Claiming something is rational does demand evidence. Logic is rigorously defined and Quantitative and verifiable.
Some things can be both, but what is rational is not always intuitive and what is intuitive is not always rational.
These properties only have an incidental relationship.