So I got to looking, and there are actually several musical versions of Batman on YouTube.
From "Batman: Beyond". An aged Batman attends a performance of a musical based on his life.
Batman does the Batusi
Batman sings "Am I Blue"
Here's a whole site dedicated to a Batman Broadway show that almost happened. Seriously, this had Tim Burton and other big name people attached.
Which may have been canceled in the wake of this MadTV skit, which in turn was a direct response to where Warner Bros. and Joel Schumacher's inability to reconcile their ideas regarding Batman. See "Batman and Robin" for how that worked out.