That seems difficult, but maybe possible to route around much of the time.
Who is this a problem for and why?
In our culture, non-binary genders are usually poorly recognised. That means any attempt to express one's non-binary gender is already conspicuous, awkward, and often ridiculed. In English, this is strongly alleviated by the existence of the pronoun "they", which has a very long history as a gender-neutral singular pronoun. "This is Sam. They work here." will unsettle some speakers, but far fewer than "This is Sam. Per works here." will.
As a result, non-binary people almost always accept the pronoun "they". I've asked around and found the following patterns. (Sorry, no linkable source; those were personal interviews.)
- Most non-binary people accept any gender-neutral pronoun. They usually suggest a pronoun (most often "they", sometimes "zie") but are just as happy with others. In such cases, your hypothetical moderator could use their favourite set of gender-neutral pronouns.
- Some people are uncomfortable with any pronoun and want their name used every time. (Usually these are people in the process of figuring out their gender.) Again, no problem.
- Some people accept only "they" and are put off by all neopronouns, because they are so intrusive and attract so much mockery. This is the case that poses a problem here.
- No non-binary people I've found accept only some pronoun other than "they". I'm sure they exist, but they seem to be very rare. This is why it's so annoying the problem is with "they" and not, say, "fae": it affects far more people.
Is this a CoC violation?
Obviously yes. It says "use stated pronouns when known". Outright refusal to do so, when one needs to refer to someone, is a clear violation of that.
Can this moderator avoid the user?
Well, isn't that kind of a jerk thing to do? Refusing to talk to or about someone because of who they are?
This is a bigger problem for a moderator, whose job is to serve the site by helping users, and who is refusing to do so for a particular user on grounds of gender.
Can calling in a different moderator help?
It alleviates the problem somewhat: moderation is getting done. But it's still, y'know, a mod giving a user the cold shoulder because of their gender.
Is there a middle ground?
Users who state pronouns in their profile can help by specifying whether alternatives are okay, especially if the primary set is gender-neutral. Rather than "they/them", say either "they/them, or your favourite set of gender-neutral pronouns" or "they/them pronouns only, please". Since (unless SE demographics are very different) most non-binary people are in fact okay with alternatives, this reduces the number of cases where the problem comes up.
Unfortunately, there is very little the moderator can do. Even if they attempt to be inconspicuous about it, by writing around pronouns or avoiding these situations entirely, the assumption of good faith is gone.
Inevitably, the refusal to use "they" will follow the usual pattern: It'll look like a subtle sign of anti-trans beliefs, making trans users uncomfortable and emboldening openly anti-trans users. Other users who have not been quite so burnt by this will be confused, say the behaviour is entirely reasonable, and make trans users even more vulnerable by dismissing their concerns. This cycle has played out multiple times so far and there doesn't seem to be a fix.