This is an interesting case, and an interesting issue. Note that the Trump case was at the federal District Court level, and the Hogan case was settled, so there was no court opinion at all. As far as I know, this issue has not been passed upon by any US Court of Appeals or Supreme Court decision, so we don't know how the courts will eventually rule.
A key question is whether the social media page in question is in fact a public forum. If it is, then people cannot be prevented from using it by the government based on the viewpoint that they express. In Trump's case, he frequently uses Twitter to announce policy decisions, which strengthens the case that his page has become a public forum. The mayor does not seem to be doing that sort of thing.
There is also the question of whether banning individuals from a Facebook page, or deleting posts to that page, are official governmental actions, and so count as "state action". If they do not, the 14th amendment, and through it the 1st, does not apply. The linked document from the plaintiff's lawyer does not seem to address this point. A court opinion probably will, one way or another.
The citation of the prior restraint cases, particularly Near vs Minnesota seems inappropriate to me. in Near the plaintiff was prohibited by injunction from publishing content in any forum at all, subject to being jailed for contempt of court should he have defied the order. Physical copies of his publication were seized and destroyed. Here Willmeng would be free to place his comments on his own Facebook web profile, or on another web site. Willmeng is merely prevented from placing them on the page organized by Berg. That is a very different thing.
Also, the retaliation cases cited all deal with official acts that are taken in retaliation for previous speech or other protected actions. Here no official act of retaliation has been identified by Berg.
The complaint says that the mayor was acting "Under color of law". This is a technical term which indicates that the act was done using official authority that has at least the pretense of being authorized by law. When a police officer arrests someone, s/he does so under color of law, even if the arrest is later held to be unlawful, because the officer depends on the authority granted by the law, and by his or her official position under the law, to make the action effective, or to deter any resistance to the action. Here the mayor does not seem to have used any official powers or status, and so I am not clear in what way those acts were "under color of law". This is important because only acts done under color of law can be the subject of section 1983 civil rights suits.
In short, I think the case as presented in the complaint is rather weak, and it is hard to say how it will be treated by a court. Eventually this issue will probably be decided by the US Supreme Court, but perhaps not in this case.
Oh, I should note that whether the poster is a constituent or not is probably not relevant. I have a First Amendment right to comment on the actions of a Mayor in Arizona, if I so choose, even though I have never lived there.