This might help: I just watched The Outlaws Is Coming for the weekly reviews, and the gag is exactly like Metaldams describes it, verbatim. There is no further action with the Japanese Beatle, and no further Beatle references in the movie.
NOW: I'm not exactly sure where it is, but in some reference work, maybe even on this site, there is a color still of the Stooges working with Danny Thomas, taken from a TV special called, I believe, either Danny Thomas Meets the Comedians or Danny Thomas and the Comedians. At one point, the stooges are cutting up, and to get their attention Danny yells at Moe " Hey ! Aged Beatle ! " ( he might have said "ancient Beatle", I'm not sure ) and Moe reacts big. Again, no further Beatles references, and the bit, whatever it was, proceeds from there. I have no idea if this video exists in any form today, but I saw it on the night it first ran in the late '60's, and that was the gag. Ask me why I remember it today, I have no idea, aside from the fact that I was a stooge freak then as well. In any case, this sounds more like the gag that Mr. Dip is remembering. I'd call you Dom, but I don't want to be getting too familiar.