There is an interesting COPE case "Author of rejected paper publicly names & criticises peer reviewer"
https://publicationethics.org/case/author-rejected-paper-publicly-names-and-criticises-peer-reviewer
I have a dilemma. A journal has invited a non-researcher, a member of an advocacy group to review our paper in which, among other things, we criticize this person's "publication" because it contained clearly incorrect information. This reviewer has focused only on defending the "publication" and not provided a single suggestion about how we could improve our paper. The paper was rejected although the other reviewer provided several suggestions about how to improve the paper, which we were willing to do.
We appealed to the journal and they response was that they don't agree that the reviewer has a serious conflict of interest.
We strongly disagree. What should I do now? Can I do anything about it? I want to make this public but this means I must reveal the name of the reviewer. The journal explicitly forbids that. Does anyone have any advice?