In Japan, the game is known as "Mad City" and was released on the Famicom system in 1988.
The game is a bit different than the US version. It's easier, some bad guys wear different-colored clothing, and Game A and Game B are switched. But if you think Bayou Billy is too hard, play Mad City instead!
As for where you can get the game, you'll have to import it. When I find out where, I'll put the info here.
If anyone has more info on Mad City, including if a soundtrack was released in Japan, please e-mail me!