"Joking" about a medical condition (particularly if this is USA) is serious.
Conceivably, it could lead to action against the company.
(If you think about it, it's one step from joking about - say - anxiety, or another mental health issue.)
I think on balance, since that's the case, really you should just firmly explain it to a manager or HR.
Anyone who doesn't understand that psoriasis is a serious, whole-life issue - the definition of a non-laughing matter - quite simply needs to be taken aside by an authority figure and have it explained.
BTW. Say we "set aside" the fact that John is millimeters away from joking about a mental health issue. So John is "just" teasing about clothes etc to juniors below John. This is totally, completely unacceptable in a professional workplace.
Again, really a more senior experienced professional needs to take John aside and explain things.
Note that,
"[John said] he just “couldn’t help it” and that “it’s just the way he is”. He insisted that he "thought fondly of me" .. etc
this sort of psychobabble from John indicates that John is, very simply, totally immature. He needs a talk from more experienced professionals.