The Joker is Probably Completely Sane and Just Uses his Public Image of Being Insane to Keep Others Underestimating Him as Just Some Lunatic
The man regularly outwits far more supposedly cognizant opponents including Batman and Lex Luthor, who are canonically recognized as two of thr smartest people on the entire planet.
Edit: I'm not saying insane people are stupid, I'm saying that Joker's mental illnesses are pretty obviously behavioral, they don't affect his perception of reality. He's perfectly capable of understanding what he's doing and how it's wrong, in fact his character almost doesn't work if he doesn't, he just thinks that it's all hilarious anyways. That's why I said he's not insane, he just pretends to be, because he'd be fit to stand criminal trial as fully competent and cognizant of his actions.
Eh, I can see where you're coming from. It's an interesting shower thought.
It just doesn't hold up to canon at all, nor casual examination of the major versions of the Joker.
The possible exception would be the 60s TV version. The flaw in that is that he never comes out on top in any way, he isn't showing signs of actually using his reputation as a weapon that helps him. But if you look at it from the outside, the character is most likely not "crazy", just playing the game of costumed criminals to the hilt. So it kinda fits because we all know that no villain on that show had a chance because the writers and production team wanted the heroes to win every time.
In comics, the older versions of the Joker were more criminal than crazy, but still showed definite signs of the crazy being there, and causing his downfall within the stories as they were told.
Being real, there isn't a single Joker. The various eras meant that social mores limited how he could be written. This was so pervasive that the characters are essentially not the same person other than general appearance and being criminals in some way, with Batman as their opponent.
You gotta understand, after a certain point (I think it was in the eighties, but opinions differ), each successive editor in chief (the equivalent of a show runner or whatever) ultimately decided not only the overall tone of the publisher's titles, but often managed the limits the company was allowed to go to.
If you look at something like the death of Robin (Jason Todd), or the premise of the Injustice "universe", that's what the Joker would be in any semi realistic sense. He's not a criminal genius, he's just a psychopath with violent and homicidal urges.
There's also the whole "agent of chaos" aspect that gets used sometimes, and the Heath Ledger version is a fairly believable take on that without being too comic-booky. But again, that's still an artifact of the limits placed by whoever has a final say. When it comes right down to it, the Joker being "sane" simply isn't a useful story for DC past or present.
All of that comic geek bullshit aside, I would actually love to see an elseworlds story with your idea. Hell, I wish I could write it and get away with it. They'd never hire my ass to do it, and I ain't fucking with copyright and trademark lawyers on retainer, but I could see how the story could be done.
Even with that, though, it would HAVE to end with the Joker losing. It would never be accepted by the public as a majority otherwise. If I was writing it, though, that loss would be him finally on death row because a single error convinced the right people that he wasn't criminally insane. No more Arkham, just solitary confinement and a drip from an IV.
That would be the final panel, too. His arm, white from the chemicals that changed him, with a needle inserted, and nothing but the typical hahahahaha that gets done when he's laughing, starting in large letters, then tapering off to ellipsis as he fades to black.
My point is that he doesn't have a warped perception of reality (in the diagnosable sense), being criminally insane would require that he be incapable of understanding what he's doing and/or the reasons why doing those things are wrong.
Joker not only is perfectly capable of understanding he's doing wrong and why the things he's doing are wrong, to him that's the entire fun of the joke.
I agree though that'd be an amazing story, but what'd be even better would be if it was a story about Batman defending the joker's insanity plea because it turns out that his own no kill rule is actually a legit mental health disorder compulsion that extends to doing everything (legally) possible to prevent a jury ruling that leads to a death penalty sentence. It'd also be a pretty stunning reversal of Joker's typical "one bad day" philosophy, that Batman is sticking his neck out in the court of public opinion on the belief that even the joker can be rehabilitated.
Honestly you could probably insert them as The Buddha and Angulimala and have a fucking wicked turnaround about what redemption actually looks like, because that story doesn't just feature an absolutely heinous criminal turning his life around and in the end actually achieving enlightenment, but also him having to live with not everyone forgiving him or even owing him forgiveness because all the atonement in the world doesn't change that he still did real harm to them that can't be undone.
Depending on the version you're reading or watching, the Joker is almost always insane and a genius. Sometimes he's homicidal, and sometimes he's just mischievous, but his insanity is his superpower. It's almost like hyper awareness, like he knows he's in a comic book so he can bend the rules of his reality to fit his whims.
He gets over on Batman, and Luthor, because he is irrational and unpredictable.
Intelligence and sanity are not linked. There are lots and lots of real world examples of very insane people being very very intelligent. There's also lots and lots of in universe content that shows he really is quite insane. The sheer number of times Joker's only goal was to get Batman to kill him should be enough proof of that.
My point is that he doesn't have a warped perception of reality (in the diagnosable sense), being criminally insane would require that he be incapable of understanding what he's doing and/or the reasons why doing those things are wrong.
Joker not only is perfectly capable of understanding he's doing wrong and why the things he's doing are wrong, to him that's the entire fun of the joke.
He isn't trying to get batman to kill him because he thinks being murdered sets someone free in any sense like Victor Szazz, he's trying to get batman to kill him because he believes it'd vindicate his view that all it takes is a single bad day for someone to become the absolute worst humanity has to offer, and because he wants to leave a permanent stain on Batman as a parting FU.
There's this really entertaining forensic psychiatrist named Eric Bender who does multiple analyses videos of pop culture "insane" characters using real scientific/medical knowledge.
Here he talks about Ledger's Joker in Dark Knight. Summary = the Joker is psychopathic (violent, doesn't care about others, others are pawns for his "game"), but not psychotic (experiencing a break from reality, hearing voices, etc). Ledger's Joker would not qualify for an insanity defense because he knows what he's doing is "wrong" and experiences the real world as opposed to hallucinations. He's not mentally healthy or normal though and possibly doesn't understand how to interact with the world.
Here he talks about Phoenix's Joker from "Joker". Summary = much the same as Ledger, but still interesting to watch. He discusses risk factors (abuse, neglect, genetics) that can be an extremely rare "Pathway to Violence" that could apply to the Joker in this movie.
So the movie versions of Joker aren't insane in the sense that they're lost in hallucinations and unable to discern reality from make-believe. The Joker likely wouldn't be sent to an asylum in the real world. The Jokers of the movies are psychopathic, evil, uncaring, manipulative, etc. For the most part, to them the world is a game and it's all about the Joker being the main character with everyone else being pawns to be used for evil amusement.
Well I mean that not how the writers who control the story write it but as far as realism there is a fair amount of mental illness that actually correlates with higher intelligence rather than the opposite. One of the hard parts of dealing with schizophrenia is they can be highly intelligent and manipulative despite their crossed wires reality wise.