Or better yet, having Dent just PAINT one side of his face.
Joker's white skin and Dent's facial scarring are crucial characteristics. Without them, the characters are just not the same.
Again, not the same. Let's look at the wikipedia introductions for each character, out of interest:
THE JOKER
The Joker is a comic book supervillain in the DC Comics universe. He is widely considered to be Batman's archenemy. Created by Bill Finger and Bob Kane, with contributions by artist Jerry Robinson, the Joker's first appearance was in Batman #1 (Spring 1940).
The Joker is a master criminal with a clown-like appearance. Initially portrayed as a violent sociopath who murders people for his own amusement, the Joker, later in the 1940s, began to be written as a goofy trickster-thief. That characterization continued through the late 1950s and 1960s before the character became again depicted as a vicious killer who carves smiles into his victims. The Joker has been responsible for numerous tragedies in Batman's life, including the paralysis of Barbara Gordon (Batgirl/Oracle) and the murders of Jason Todd (the second Robin) and Jim Gordon's second wife Sarah Essen. He has been described by the Martian Manhunter in the JLA series as "the most corrupted of human souls."
Interpretations of the Joker in other media include Cesar Romero's in the 1960s Batman television series, Jack Nicholson's in Tim Burton's Batman, and Mark Hamill's in Batman: The Animated Series and other DC Animated Universe shows. Wizard magazine listed him the #1 villain of all time in 2006. As played by Nicholson, The Joker ranks #45 in the American Film Institute's list of the top 50 film villains of all time. Heath Ledger will portray the Joker for director Christopher Nolan's Batman Begins sequel, The Dark Knight.
TWO-FACE
wo-Face is a fictional character, a supervillain and enemy of Batman in the DC Comics Universe. Created by Bob Kane and Bill Finger, he first appeared in Detective Comics #66 (August 1942).
Bob Kane was inspired by a movie poster advertising the Spencer Tracy film Dr. Jekyll and Mr. Hyde and conceived the idea of a villain with a dual personality. Two-Face was once Harvey Dent, District Attorney of Gotham City and close ally of Batman. After a criminal disfigured half of his face with acid, Dent became the insane crime boss Two-Face who would choose to do either good or evil depending upon the results of flipping a coin — a device which was taken from the 1932 version of Scarface.
Originally, he was one of many gimmick-focused comic book villains, plotting crimes based around the number two, such as robbing Gotham Second National Bank at 2:00 on February 2. In recent years, writers have portrayed his obsession with duality and his criminal behavior as the result of multiple personality disorder and a history of child abuse. He obsessively makes all important decisions by flipping a two-headed coin, one side of which is scratched.
Although too gruesome for the 1960s television show that popularized Batman and much of his rogues gallery, Two-Face has been a prominent foe of the Dark Knight and was played by Tommy Lee Jones in the 1995 film Batman Forever. The Harvey Dent character will also play a part in the upcoming film The Dark Knight, portrayed by Aaron Eckhart.
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Other than his "clown-like appearance" (which The Joker still has in "The Dark Knight", regardless of how he gets it), no mention is made of his bleached skin in this introduction, the aim of which is to summarise the core elements of the character. However, in the introduction for Two-Face, we go into detail about his scarring, and what it was inspired by, and what it has caused. It is crucial to his character. He's even named Two-Face, for literally having "two faces".
We know that the scarring is at the core of Dent's breakdown. But we know nothing for sure about The Joker, and his madness.