I feel like Two-Face having a third split personality makes some amount of sense, although I also get why it isn't explored that often. I mean… Two-Face is known for Twos. His gimmick is duality. Having a character like The Judge ruins that gimmick. But at the very least, I do like when media with Two-Face tries to add elements of that third personality into their stories.
For an example of that, I love pointing at the Telltale Series games. When you get the choice to save Catwoman from a sneak attack by some crooks or save Harvey from a giant spotlight crashing onto his skull, the game then offers a branching path for Harvey. Both paths end the same way (because he'd already been injected with a serum by the children of arkham that lowers impulse control and makes people more prone to strong emotional responses), but they develop his character in significantly different ways. I suppose I should be worried about spoiling this for people, but I personally like spoilers, so i'm not that bothered.
If Batman saves Harvey, then the drug still works it's magic and makes him a paranoid wreck, obsessed with consolidating power around himself. But it also makes him less bitter towards Batman. His darker side begins to idolize the vigilante justice aspect of Batman, and I believe he even creates a special police force modeled after Batman (I think). In essence, he becomes a fanatic obsessed with the strength of the bat. Elements of The Judge can be seen in how his paranoia makes him try to become a dictator.
So Two-Face without the scars is (if i remember correctly) a lighter shade of evil. He hasn't got the physical trauma to deal with, he's only got the mental stuff. And it's a little easier for him to manage one without the other. He still looks the same, so it's not like he's gotta be scared of being judged for his looks. So that might honestly help his sanity a tiny bit. I remember the scene where you see him in his office before his mayoral portrait is painted. Without the scars, he's paranoid and messing with his coin, but he's not angry and violent. He's more just paranoid and lashing out.
Meanwhile, a Batman who lets Harvey's skin burn and scar over ends up creating the Two-Face we all know. This man spends half the game hiding behind bandages, or phantom of the opera style face masks. He resents Batman, since Batman saved Catwoman and left him to burn! He can't even have a normal mayoral portrait taken, because one side of his face is red and covered in welts and scars! This Two-Face is so deranged that he can't even be reasoned with. Which kind of makes sense. I imagine it's hard to think when your skin is always pained and your mind is full of a drugged substance.
In either case, Harvey Dent gets put through the ringer. He gets drugged. He thinks his girlfriend cheated on him with his best friend. He potentially mauls a criminal to death with his bare hands. He has a psychotic break. But there's two different flavours of psychotic break. There's the version where he's obsessed with bringing batman style justice to the city, and then there's the version where he hates Batman and blames him for Gotham becoming a hellhole that he has to fix.
If you read into all this, it really feels like Harvey develops a different split personality depending on what happens to him. If he isn't scarred, he gains a split personality in the form of "The Judge", an authoritarian obsessed with law and order (at all costs). And if he is scarred, then he gains a split personality in the form of "Two-Face", an angry and violent man who thinks he has to burn the city to the ground to make it better. Both his negative personalities showcase his obsession with justice and order, but they showcase it in slightly different ways. It's pretty cool.
