What melody note (or notes) does the tune begin and end on? Sometimes that's the best way to start figuring out less obvious keys and progressions, working backwards to figure out the sense of the harmonic structure from there.
A few years ago I had to transpose "Lucy In The Sky With Diamonds" two steps lower to accommodate the baritone voice of one of my bandmates, and that's how I worked it out. Not exactly a I-IV-V song there!
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