I don't know how helpful this will be, but I'll try. It appears (as will probably come as no news flash to you) that there is no mp3 of it available anywhere. There are no albums at all by Hairbrain listed on Amazon. There is no entry for Hairbrain on allmusic. Nothing on Amazon is a pretty good indicator that they never had an album out.
On IMDB, it lists soundtrack info and lists the song being written by Jim Wirt and, as you know, being performed by Hairbrain. My guess - and this really is a total guess - is that Hairbrain is what he called the group of people that he got together to record the song that he wrote. Still guessing here, but I wouldn't be surprised if there is no band. He may have written the song for the soundtrack and had studio musicians play the music he wrote in order to record it. If that were the case, well you need some name to list on the soundtrack, ya know? So instead of listing just his name, he might have basically made up a band name. This is all guesswork on my part but I have read similar stories in the past.
Also....I don't know if you've heard a full version of this song or not, but sometimes in cases like this, there isn't one. Sometimes they just record the part of the song that they know they need for the movie and leave it at that, because they know it's not going onto any album later or anything, and they aren't really thinking so much about us obsessive fans :( (Plus the studio time is absurdly expensive - real tough to justify if finishing the song is in effect for nothing.) A really good recent example of this is the cover of Scandal featuring Patty Smyth's "Goodbye to You" that was in a Cotton commercial and a whole bunch of people were real desperate to get there hands on it. Well it turned out that there was no finished version of the song - the cover artist, Lizzy Seich, came onto the youtube page of the commercial and explained the situation herself.
If you really, really, really want to know, try contacting Wiltone Records (from the IMDB listing) or the song writer himself, if you can find anyway to get a message to him. (Probably not easy, of course.)
Sorry that that is clearly not what you want to hear....not to mention that some of it is guesswork....but for some reason I searched extensively and this was my conclusion/best guess.
P.S. Is this movie any good?