What we can tell you ... (1) For trademark purposes, there is no discernible difference between "Bat Man" and "Batman" especially for radio listeners. Having a soundalike mark doesn't mean you can't use your name, but soundalikes are often considered as substantially similar or identical. (2) DC seeks a broad range of protection for its marks and logos (and there is a wide variety of Batman logos). DC has registered numerous trademarks including, obviously the word, BATMAN, for entertainment services. (3) DC Comics spends a lot of time and money going after anyone who encroaches their turf -- they even stopped a comic book store using 'Batcave' (at least that's what we said in one of our books).
Bottom line dept. Since you are offering entertainment/educational services and have an identical mark, DC would have a strong case against you. So, if you're looking for a moniker you can count on for a long term success, we wouldn't bet on Bat Man.