In the video in post 3 above, Peter Tork is playing a Vox organ... but that's just a prop. I believe the organ used in the recording was a Hammond. For the live performance in post #4 (of a different song), they're using a different sound, though it happens that it is also a Hammond organ sound, though from what little I can see of the gear. it wasn't necessarily a Hammond organ, it could have been some other keyboard emulating a Hammond organ. There may be some irony here in that it may be that the "real" Hammond organ used on the Believer recording is less immediately identifiable as a Hammond than is whatever keyboard is being used in the live performance to emulate a real Hammond organ. ;-)
Now to the question at hand, of whether you can get something close to that particular organ sound out of the MX88, I don't know. But heck, the MX88 only has 42 sounds in it, and that opening part is only a few seconds long, so it would only take you a few minutes to play the part with all 42 organ sounds and see if you find one that sounds right to you.