The Batman of Earth-Two was supposedly the 'alternate reality, do-gooding, cool in tights, anti-Joker, manly and muscle-bulging, better-than-anyone-else, self proclaimed real version of the urban terrorist Bateman come to our reality. He spent most of his time gunning down the homeless of Gotham City with a police issue revolver and leaving dime store robbers with permanent injuries. Unfortunately, he was never noticed by the real Bateman or even the Joker.
As a result, The Batman of Earth-Two turned to selling his gun and ammunition for money to spend on expensive beer and doughnuts. He was last seen drinking at a bar frequented by Killer Croc, the Riddler, and Amygdala. He fared rather poorly against the Cookie Monster in a poker game, after unwisely challenging the member of the Sesame Street Mob to bets after watching him beat Killer Croc. The Batman of Earth-Two then vented his rage by thrashing the bar and beating up Captain Pathetic, a minor costumed crook at the counter.
The Batman of Earth-Two was then seen briefly advertising a line of soft drinks in an unsuccessful attempt to combat the 'Zesti Soda' monopoly in South America. He ended up wandering amongst Gotham's streets picking fights with muggers. After being beaten up by several homeless men who took his money and his costume, the Batman of Earth-Two burst into a meeting the Joker was having with the Arkham Asylum staff to confront his self-proclaimed nemesis once and for all.
Unfortunately, the Joker only took 40 seconds pause from the meeting to show the Batman of Earth-Two the best way to sharpen his pencil.
The self-proclaimed 'Batman of Earth-Two' will forever be remembered for being more manly, cool, and macho than the Bateman of our reality. He also upheld the policy that "Killing is fun", which was one reason he was able to separate himself from the wimpier Bateman.