Burton was a huge Prince fan. And around the time when Batman was being made, Prince was uber dark.
Lovesexy/The Black album highlight this. Princes original material for Batman was much much less commercial.
I have some of it.
Tracks like "Dance with The Devil", the original version of "Rave into the joy fantastic" as well as the original version of "Electric Chair", "200 Balloons" (Joker's Theme) and nearly 10 minute dialogue version of "Batdance" were massively more interesting than what ended up in the movie.
It was Warner's and the producers who wanted something more commercial, and well they got it. The Album and singles were a huge part of the PR push, well they were in Europe anyway, where Prince was still massive.
In regards to whether his music was tonally correct or not, well Batman was supposed to be timeless, neither period or futuristic, which is why (at the time anyway) using something very modern was a good way to go. Fedora's, period costumes mixed with modern hookers on the street and the synth beats of "electric chair" playing in the background was pretty cool IMO. It may not have dated that well, but I still dig it.
Prince made the BATMAN album his own thing too. It's really about the duality of dark and light. The liner notes are very interesting. He was basically breaking Bruce Wayne, Batman, and a third part of the characters psyche (Gemini) into separate characters who spend most of the album conversing, arguing with each other about right and wrong and society.
It's a hell of a lot different and darker that what's in the movie, because Prince was big into Batman. Which is why his video's
have Prince (Gemini) playing both good and bad. Batman and Joker. There.. how much of a Prince nerd am I?
Some of the music is awkwardly wedged into the movie though.