Some people "prefer to use only their name and not use pronouns at all". If that is all that is requested, it should not be counted as "obviously unreasonable" and "requested unkindly". (See point 8 of the implementation FAQ).
None of: They, he, she, and definitely not it
Cody tells us the above was removed from your profile by a Stack Exchange staff member. So in this case, you might consider using the "Contact" link at the bottom of the page, to submit a request to the staff.
I think the FAQ tries to focus on behaviour. In some cases, perhaps a staff member would look at a user's behaviour and judge a request in their profile as being "obviously unreasonable" or "trolling". I have no insight into whether that was the case for you.
I'm kinda anxious. If I was you, I would probably try to make sure I was being as convincing as possible. Considering the current unpleasant events. So I have some further thoughts.
In future, it might be useful to include a link to the Wikipedia section, or a similar introductory resource. And the quotation from it, or something similar.
Particularly because this specific possibility has not been mentioned in the FAQ. Perhaps an oversight in the FAQ. Although I would hope a moderator would be able to realize that this could be a possibility. Note that in some surveys, this possibility is no less popular than the most popular neo-pronoun (about 10% of non-binary survey takers).
It might also be useful to drop the "Peilonrayzelf" line. It's a fun line, but I don't think it is necessary. If it's the equivalent of "myself", then you're the only person who needs to use it. (Deleted, see comments).
"considered to be trolling behavior" ... a "protest against site policy".
I have seen a number of Meta posts, about how users should engage if they suspect trolling. Here are links to two that might be relevant.
Neopronouns or trolling? (answer by Shog9)
How should we respond to posts that might be written by trolls? (answer by Kate Gregory)
"Please don't call someone's pronouns made up"
If a moderator needs to publicly explain a decision, they should try to exercise all reasonable care.
I note the FAQ does not use the phrase "made up". For a new word to enter the language, the first thing that has to happen is for someone to make it up.
There have been multiple examples of people complaining in a hostile tone, that all neopronouns are "made up". Part of the hostile tone comes from the connotations of that phrase, that they're not "legitimate".
Considering all of the above, I would hope moderators would be careful with this phrase.