Since I remembered seeing references to Baron von Blubba as the main villain, I figured that's who this was. But no, the Baron is actually the whale skeleton (also known as the Skela-Monsta) who shows up when you've been on a single level for too long. (He must have attended the Evil Otto School of Villainy.)
No, the name for this boss character is apparently "Super Drunk" (no relation to the Bruce McCulloch character, as far as I know). He's the larger version of the bottle-throwing enemy known in the original Japanese simply as a "Drunk," but translated to "Willy Whistle" by possibly teetotal Americans. The Super Drunk is also called "Grumple Gromit," but he's Super Drunk in the credits, and it seems to be the favored name online by far.
Incidentally, I'm pretty sure that the Super Drunk's stage is the only one that doesn't play the theme music, which has to be right up there with Super Mario Bros. and Zelda as one of the catchiest video game themes. And it's a good thing it IS so catchy, considering how much you'll hear it. It doesn't even stop in between stages, although it does temporarily halt for some events (like the arrival of the Baron, for instance).
Bubble Bobble theme
Here are a few other links that I found regarding the game:
The best ending (Hey, if the dragons turn into those wide-eyed youths at the end of this game, how come they're dragons again in later ones?)
Review of the game
Guide to every level
Drew Mackie recounts how the cover art inspired a nightmare.