I'm interested in what you mean by a moral rule/norm/fact working universally.
Personally, I am a moral rationalist. Ordinarily, that would entail "moral realism," but for me it's more of a form of moral constructivism, like mathematics. When we define our fundamental moral terms, we can extrapolate true moral statements, but this is mostly an exercise in technical language. We aren't really describing "reality" when we do this.
My indirect realism is similarly semantic. I'm aware that the sensations that I experience are external to my control, and they consist of a a self-consistent data set, which in modal logic we refer to as the "actual world." So I'm okay with describing that as an external world and that's what I call "reality." I would never use "real" to refer to abstract constructs like morality.
Nonetheless, I do regard some moral facts to be objectively true and knowable a priori by the intellect. Part of this is because I view morality as dealing with normative statements, which have an inherent drive to them. Rationality itself has an inherent drive to its process.
This means that I am opposed to moral sentimentalists, who instead rely on emotions like empathy, love, or compassion (and sometimes disgust, anger, shame, guilt, etc.) as the main drives behind morality. This is mostly due to the fact that I'm alexithymic and struggle with dissociation, so I'm completely out of touch with my emotions and I can't build a philosophy from them. I have to default to rationalism.
We've discussed in the past how objectives are subjective, but rationality itself is subjective. It's something subjective agents experience; we all have our own rational drives and rational intuition. It's "universal" in the sense that everyone has these drives and intuitions to some degree, although it takes study and practice to refine them. You can have rational approaches to your own subjective, internal states.
You can theoretically live a life of pure rationality. It's difficult in practice. I know that I fall short sometimes, but I'm a lot further along in cultivating the discipline and understanding necessary to do this than I was a few years ago. To me, this quest takes on a moral connotation regarding the integrity of my character.
All of this said, I recognize that there are other definitions of morality that extrapolate from different sources. You could say that which one we choose is subjective. I wouldn't go that far. I don't think I can demonstrate that any other axiomatic definition of morality is necessarily false, but I can demonstrate that my morality is a consequence of rationality. In that sense, it is true in the same sense that logical conclusions are true, because formal logic is just a formalization of the rational intuition. "Logic" in its broadest sense just referred to rationality and reason in general before we formalized it.
So my moral position is logical. Other positions might be, too, and they might even be equally logical, but that is subjectively not important to me. I'm okay with there being numerous different meanings for "morality" depending on the context. All words have that