In all these years, and all the CDs I bought, I've never found a CD release that had the actual theme used in the TV series. I checked here, and they all have some variation of it. I'm looking for the music used in the show itself, from the spinning dashboard onward. Does that even exist on CD?

Like this: https://www.youtube.com/watch?v=1jgE-lrfZ3k

Probably not. You'd most likely have to rip it and deal with dialogue and fx.....

There was a bluray release last month.

http://www.blu-ray.com/movies/Batman-The-Complete-Series-Blu-ray/103989

However, while it is all bluray, the audio is just:

English: Dolby Digital Mono

So dialogue and sfx will have to go with the territory.

Even if you were to find a DVD set in stereo, it's most likely a mono track doubled to fit a stereo track.
The original audio for the show is in mono:

http://www.imdb.com/title/tt0059968/technical

The LaLa Land release for the batman movie has a pretty close version of the theme, TV size.
(So does the FSM version.)

The end credits are on one of the TeeVeeToons collections I own. And the season one version of the theme has no FX on it so I could rip it for you.

Season 3 has Batgirl's kick and Batgirl-cycle FX over it.