You made the judgment that somebody is a man hater. You made that by a set of standards that you had. In saying it, you presume two things:
1) Your standards are valid. They may be, but it's very hard to demonstrate to everyone's satisfaction. People are more likely to presume that you're just trying to insult. See below.
2) Whether valid or not, calling them man haters is either counterproductive, insulting or both. Let's say your standards are valid. You're either dealing with somebody who is consciously hateful. If the term "man hater" is supposed to shock them out of hating men, it won't work because they already know.
More than likely, however, you have somebody who's unconsciously hateful. By definition, the unconscious is something that a person cannot become conscious of, at least, not without years of therapy. You can't provide them with years of therapy in a board discussion. If you call them man haters, they will sincerely deny it. Even if you have overwhelming evidence, even if everybody else sees it. (Truth is, due to our unconscious, other people can see things about you that you can't see yourself. The unconscious gives you blind spots.) So, they could only see "man hater" only as an insult, and others are likely to side with them because they will likely give them the benefit of the doubt and believe alternate explanations in the face of such an extreme accusation. You don't have a chance to present your proof.
My advice: you shouldn't try to argue or even acknowledge people's unconscious motivations for anything. Instead, take apart the conscious arguments. This disarms the unconscious. You can't change it.
Save "man hater" as an insult to people who are deliberate and conscious of it, and keep in mind you won't run into many. Also expect it won't shock them. It won't be a revelation, and it definitely won't make them change.