Here are the main sources I've used:
Bacon, T. 2006. "Paul McCartney Playing the Great Beatles Basslines," Backbeat Books.
Dowlding, W.J. 1989. "Beatlesongs," Fireside/Simon & Schuster, Inc., New York.
Everett, W. 1999. "The Beatles as Musicians, Revolver through the Anthology,"Oxford University Press, New York, Oxford.
Lewisohn, M. 1988. "The Complete Beatles Recording Sessions," The Hamlyn Publishing Group Limited, a Division of The Octopus Publishing Group plc, London.
Rolling Stone. 2010. "The Beatles 100 Greatest Songs," Special Collector's Edition. Jann S. Wenner, Editor and Publisher.
Shea, S. and R. Rodriguez. 2007. "Fab Four FAQ," Hal Leonard Books, New York.
I've also consulted Andy Babiuk's book "Beatles Gear," although that doesn't tie the instruments into specific songs the way some of the other references do.
Of those I've listed, they all seem to have inconsistencies even within themselves, and certainly between themselves. I put the most creedence in Everett, Lewisohn and the Rolling Stone article (I actually found this at Hartsfield-Jackson International Airport in Atlanta while I was deeply immersed in this research). Shea and Rodriguez seem pretty good as well (based on what I thought I already knew), but they don't provide any references to their statements. Dowlding is often pretty far off (but cites various good references such as interviews with John, Paul and George for several of his more accurate claims). Bacon is a transcription of several of Paul's Bass parts, so it has some good info, but it is limited in this application because it is only for tunes Paul played bass on.
Walter Everett claims that John used the Fender Jazz bass on Helter Skelter. Lewisohn says John played, but doesn't specify the bass. Bacon says it's either John or Paul on Bass VI (???) and Dowlding says it's John and Paul (I may believe that!) but doesn't specify the bass. Babiuk, however, indicates that the right-handed Jazz arrived at the beginning of '69.
So, we basically have Everett vs. Babuik, w/ some indication of John on Bass VI. I coupled this available information with my ears and what I know about the Bass VI (it is a fairly mellow instrument; more like a baritone guitar than a bass) and concluded that it's John on a RH Jazz. Then again, would Lennon have felt commfortable doing those fairly quick descending runs on what would seem like a very widely-spaced neck on what also would have been a new instrument to him? Maybe not.
You know, maybe. Maybe not. I will say that it was just tonight that I saw in Babuik's book that the RH Jazz apparently/allegedly arrived in early '69. So, we have to decide if Babiuk is correct or if Everett is correct. This is the sort of problem I typically run into. I consider both Everett and Babiuk to be pretty well-researched and reliable sources. But one of them is wrong. I was just trying to explain how I had arrived at the conclusion I had posted earlier today. Now, I'm not so sure. But, my ears tell me it's a Jazz. And I'm 99.9999% sure it's John.