Anyway, I have an idea how you could fake it on stage pretty well, hear me out. So the sound has 3 components: a hammond organ, through a resonant filter sweep (driven manually or by a very slow LFO), than a square-wave AMP LFO timed to make the eighth-notes.
The first two stages are easy, just pull up a good hammond patch, create a filter sweep, and modulate it with a very slow LFO. For a quick-and-dirty, you could just stop there and play the eighth notes, which would probably satisfy most shows, though it won't be 100% authentic, and maybe a bit tiring after a while.
If you want to put in the amp LFO but not have to use a click, here's what I've done with similar things in the past. Most synths have a monophonic LFO trigger option, so that all notes are modulated by the same LFO timing, but not a "sync" LFO which is constantly driven by a timer (you'd need to give the drummer a click to that... and then... drugs). The nice thing about this is that as long as one key is still being held, the LFO produces exactly the same timing, so you can go a few measures with the robotic pulses, and slightly slippery chord changes as long as one note remains the same (which in that song, many are). But then the moment you feel it getting slightly off, you can play the chord again, and the LFO will restart and snap to the beat you played it on.
I use this trick with an accordion patch in a Pogues song ("If I Should Fall from Grace with God"). The accordion player starts madly pumping the box and doing sixteenth-notes that work very well with a pulse LFO. So I just through on a mono-LFO, but every time I switch chords, I can sync back to the drummer, no matter how much he drifts (and he does!).
