Aristotle argues somewhere in the Nicomachean Ethics (I can't seem to find the exact reference at the moment) that it's pointless trying to teach ethics to someone who doesn't have the virtues at all. If we haven't already learned to be honest in general (and how honesty can come into conflict with other virtues and how to navigate some of the simple cases of those conflicts), then theoretical arguments about the value of honesty in a life well-lived aren't going to change our behavior. Even more — if I'm remembering the argument correctly — Aristotle thinks that we won't even really understand basic ethical concepts like honesty if we aren't already at least somewhat honest.
Aristotle's moral psychology is radically different from Kohlberg's scale. (Let me note in passing that there are some important feminist critiques of Kohlberg's scale. See this Stanford Encyclopedia article.) But if it's fair to call possessing an Aristotelean virtue "moral intelligence," then on his view ethics does require moral intelligence.
But not everyone is an Aristotelean, so here's a related perspective from my experience teaching Introduction to Philosophy. I've taught Intro at both very prestigious private universities and as night classes at community colleges. As you might expect, the students at the prestigious private universities were overwhelmingly 18-22-year-old white and wealthy or upper-middle-class children of the suburbs. And the students at the community colleges were very diverse in terms of age, race and ethnicity, and class. Many of the community college students were in their 30s or older, worked full-time, had kids, and took night classes to get a degree that would get them a raise at work.
My Intro to Philosophy course had a loose theme of deep disagreements, and so we covered a lot of controversial issues — socialism, sexism, racism, mass incarceration, climate change, vaccine skepticism, etc. The students at both kinds of schools would get engaged in our class discussions. But almost none of the privileged young students at the prestigious universities could relate personally to these issues. We were continuing intellectual debates they had in their private high schools. On the other hand, most of the disadvantaged students at the community colleges could relate personally to the issues we were talking about. The philosophers we read were providing them with concepts and theories that helped them understand the challenges they faced in their lives. (I'm a democratic socialist, so for me this was the real point of the class.)
Standpoint epistemologists (including but not limited to feminists) argue that the different social positions of knowers have different epistemic consequences. My community college students were often able to understand the philosophy they were reading better than my prestigious university students, because they had been required to learn to navigate the world described and criticized by that philosophy. That intelligence — which you might or might not want to call "moral intelligence" — definitely helped some of my students understand philosophy better.