I've never played anything like that myself, but its a damn fine idea. The Red Rain
trilogy are actually some of my favourite Batman comics (an unpopular stance, I know).
I think it would be easy enough to have a Batman very reminiscent of the Golden Age Sandman, or a bit more Doc Savage-like, and that would fit in nicely with a 1920's Cthulhu setting.
Also, there is a bit in the movie Batman Begins
, where he is swooping out of the darkness, picking off gangsters at the docks one by one and terrifying them. Its more like Alien
than Adam West. I don't see why Bats couldn't make the jump to the horror genre.