Here's a little tidbit from "The Arizona Republic", Alice's hometown newspaper:
Andrew Lloyd Webber and Tim Rice invited the rocker to record that very song for a London cast recording album.
"It was funny," Cooper says. "Because I’ve known Tim Rice, the lyricist, forever. So when they got ready to re-record the original soundtrack for 'Jesus Christ Superstar' maybe 10 years ago, he says, ‘Do you want to do the Herod song?’ Victor Spinetti did it before me. He was in ‘A Hard Day’s Night.’ A great, great actor. I said, ‘Wow. Really? I could definitely put a spin on it.’”
So if I read that right, Alice's performance made it on a cast album without really being in the cast. Nicely done indeed.