I have been an Alice Cooper listener since about 1971. Where did you hear that that song was connnected to Morrison? Seems to me it was a straight on lyrical tribute to old west gunfighters.
He mentions it quite often in interviews.
http://www.rollingstone.com/music/song-stories/desperado-alice-cooper
He used to sleep it off under Morrison’s sofa.
Jim was part of the Hollywood Vampires.