the person that peterson reminds me of most is zizek - he appeals mostly to uneducated white men who never went to university, who get some kind of ego boost from thinking that they're finally being allowed to engage in a debate that they were never previously permitted to engage in, because nobody ever thought they were intelligent enough to engage in it. but, there's essentially nothing of substance to what he's saying, and when he's not being dishonest, he's being distracting.
and, he is deeply dishonest.
the surreal thing is that you'll see him being presented on the other side of the argument - as though freud or jung should be taken more seriously than derrida or lacan, or foucault. you've just moved across the rhine. it's the same bullshit. and, the guy really belongs in the same category as much of what his proponents are trying to contrast him against.
as mentioned: i don't want to engage in this debate, because i don't think it's worth my time. other than to clarify the facts of the situation when he attempts to distort them, that is to fact check him, he's really broadly not worth responding to.
and, that is my suggestion to dealing with him - don't bother engaging him. just fact check him and leave it at that.