Spirit of the law - likely, but I feel like someone's picking a fight.
Letter of the law - yes, but it's a chilling effect on moderation.

With some timely clarification of the new CoC, we have this snippet.

M2. I'm a moderator. I often have to refer to other users, I can't choose which ones I have to refer to, and often it isn't feasible to avoid pronouns. So do I really have to use pronouns I find uncomfortable?

Yes. As a moderator, you're held to a higher standard and are expected to set a positive example for your community. Using incorrect pronouns or conspicuously avoiding using pronouns is disrespectful. Telling them you refuse to use their pronouns is rude. If you think you will find using pronouns as stated difficult, please try. Most moderators won’t have to address this frequently. If you need help or have questions, contact the CM team for guidance. After that, if you cannot in good conscience follow it, reach out so we can find a replacement for you.

In essence, this translates to:

If you have a diamond next to your name, you will use stated pronouns irrespective of how you feel about them, or you'll be replaced.

...and these are just the table stakes for being a moderator these days. Being bound to the CoC is a thing that all moderators are, and if this also includes compelled speech in this fashion, then so be it. The only real option would be to not be a moderator, and you can continue to write in a way you find comfortable.

The reason I feel like this is picking a fight is that, had no one known your pronoun, the most appropriate ways to refer to you would be either your display name or "OP". The only time then that a pronoun becomes known is when it's commented to that effect - commented being the operative phrase here - adding information which may or may not be germane to the actual discussion at hand.

I come from Stack Overflow, a land where the compiler doesn't care what state you exist in so long as you satisfy its syntactic requirements. I see discussions adding context about personal identity as noise, so I have a much starker and much more sterile view on this. Reconciling that with the rest of the network is difficult; there's no middle ground on what a moderator can or cannot do since they are meant to represent the best of the community.

I suppose it means that if you don't want to use pronouns in all cases, then you shouldn't try for modship, since that seems to be the position that the Community Team isn't going to bend on.

The use of "conspicuously" makes it a Weasel Word and basically makes it a truncheon with which to beat mods into submission.

Given the lengths to which they have already gone to enforce the policy before it was even written, there is simply no middle ground, no compromise, and nothing but unquestioning compliance.

If SE were willing to drag someone through the mud, in a very public way, commit blatant libel, and refuse to correct their mistake, all for the high crime of asking a question, where do they stop?

Can anyone honestly say that SE has not created an environment of fear?

SE did all of this to Monica, who did nothing wrong.

What will they do to someone who actually DOES violate the CoC, by accident or on principle?

I see no possibility for middle ground, only blind compliance, with serious, very public consequences for anyone who steps even slightly out of line.

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.