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David Moss mentioned a "long tradition of viewing ethical theorising (and in particular attempts to reason about morality) sceptically." Aside from Nietzsche, another very well-known proponent of this tradition is Bernard Williams. Take a look at his page in the Stanford Encyclopedia of Philosophy, and if it looks promising check out his book Ethics and the Limits of Philosophy. You might also check out his essays "Ethical Consistency" (which I haven't read; in his essay collection Problems of the Self) and "Conflicts of Values" (in Moral Luck). There are probably lots of other essays of his that are relevant that I just don't know about. Another essay you might read is Steven Lukes' "Making Sense of Moral Conflict" in his book Moral Conflict and Politics. On the question of whether there can ever be impossible moral demands (that is, situations where all of the available options are morally wrong, potentially because of conflicting moral requirements), one recent book (which I haven't read, but sounds good) is Lisa Tessman's Moral Failure: On the Impossible Demands of Morality (see also the SEP article here). Don Loeb has an essay called "Moral Incoherentism," which despite its title seems to deal with something slightly different than what you're talking about, but might still be of interest.
The piece that comes the closest to speaking directly to what you're talking about here, that I know of, is Richard Ngo's blog post "Arguments for Moral Indefinability". He also has a post on "realism about rationality" which is probably also related.
On "consistency with our intuitions," a book to check out might be Michael Huemer's Ethical Intuitionism. And of course the SEP article on ethical intuitionism. Though of course intuitionism isn't the only metaethical theory that takes consistency with our intuitions as a criterion; David Moss mentioned reflective equilibrium -- and I definitely second his recommendation to look into this further -- and Constructivism also has some of this flavor, for instance. Also check out this paper on Moorean arguments in ethics ("Moorean arguments" in reference to G.E. Moore's famous "here is one hand" argument).
David Moss also mentioned "hyper-methodism and hyper-particularism." Another paper that touches on that distinction, and on Moorean arguments (though not specifically in ethics) is Thomas Kelly's "Moorean Facts and Belief Revision."
This is a pretty standard view in philosophical ethics, reflective equillibrium.
For a somewhat opposed approach, you might examine moral particularism (opposed to moral generalism), which roughly holds that we should make moral judgments about particular cases without (necessarily) applying moral principles, so while the particularist might care about coherence in some sense (when responding to moral reasons) they needn't be concerned with ensuring coherence between moral principles and between principles and our judgments about cases. You might separately wonder about how much weight should be given to our judgements about particular cases vs our judgements about general principles on a spectrum from hyper-particularism to hyper-methodism.
In terms of other characteristics of a good ethical system, I think it's worth considering that coherence doesn't necessarily get you very far. It seems possible, in principle, to have coherent views which are very bad (of course, this is controversial, and may depend in part on empirical facts about human moral psychology, alongside conceptual truths about morality). One might think that one needs an appropriate correspondence between one's (initial) moral views and the moral facts. Separately, one might think that it is more important to cultivate appropriate kinds of moral dispositions than to have coherent views.
Related to the last point, there is a long tradition of viewing ethical theorising (and in particular attempts to reason about morality) sceptically, especially associated with Nietzsche, according to which moral reasoning is more often rationalisation for more dubious impulses, in which case, again, one might be less concerned with trying to make one's moral views coherent and more in applying some other kind of procedure (e.g. a Critical or Negative one).
There is a lot of empirical moral psychology on these questions. I'm not sure specifically what you're interested in, otherwise I would be able to make more specific suggestions.
I think more applied messaging work about the reception of EA and receptivity to different messages would also be valuable to explore this and would likely help reduce risks which EAs run when engaging in outreach or conducting activities which are going to be perceived a certain way by the world.
Could this be considered similar to the bias/variance tradeoff in machine learning?