Well, thank God for the capo! I had that flash of insight and put one on the first fret, intending to move it up until I established a key for the song. As luck would have it, it sounds in the key of Ab on my CD, which meant that playing in G with the capo on the first fret, I was able to play along with the song. Much to my surprise, the chord progression is very straightforward:
G Billy Rose was a low rider; Em Billy Rose was a night fighter C D7 Billy Rose knew trouble like the sound of his own name C D7 Busted on a drunken charge C D7 Drivin' someone else's car C D7 G The local midnight sheriff's claim to fame G Em In an Arizona jail there are some who tell the tale how C D7 Billy fought the sergeant for some milk that he demanded. C D7 Knowing they'd remain the boss C D7 Knowing he would pay the cost C D7 G They saw he was severely reprimanded Em D7 Em D7 In the blackest cell on "A" Block he hanged himself at dawn Em D7 With a note stuck to the bunk head C D7 Don't mess with me, just take me home. G D7 G Come and lay, help us lay young Billy down.
I'm sure I've missed some passing chords in the process, but this progression scans pretty well.