Yes, this is how John plays it. He didn't play it as succinctly live as he did in the studio though.
I can only imagine that as you are reaching with your little finger your index finger barring all the strings releases slightly on the fretboard and you are not pressing hard enough with your little finger. I will check again when I have a guitar in my hands. It is a difficult riff to pull off fluently time after time though.
I'm the least person to give advice on guitar playing, but what I do is, instead of going down to the E shape on third fret, I just start that part of the passage on the A string 12th fret. By fretting that far up the neck and picking in the middle of the string, you retain that Rhodes-piano-like tone that John gets. I guess that when George comes in with the lick, he's playing down the neck on the 3rd fret position (actually starting the lick on the 5th fret, D string).
Being woefully self-taught on guitar (I've been formally trained on other instruments), I never learned how to fret with my pinky. I have heard that B.B. King is the same way.
Otherwise, it appears you are playing the lick the same way I do. In the Philip McDonald book, Sean Lennon says that his dad was double-jointed, which made him REALLY GOOD at playing guitar chords.
I never knew John was double-jointed. He did seem to have fairly big hands though as when he played C7 shape chords up the neck of the guitar he could easily get his thumb right round the neck so he was barring the E and A low strings.
AND... he played it on that Gibson jumbo ac/elec. guitar. On my '78 Yamaha jumbo, the feedback sound is exactly the same and the tonal quality is the same. That 'rhodes piano' quality is just ancient acoustic pickup tech going through ancient amplifiers so it sounds nothing like a guitar.
Friend: My favorite Beatles song is Band On The Run.
Me: It's not a Beatles song.
Friend: Wot! That guy, that guy, Paul McCartney sings it right?
Friend: Alright! That one is my favorite Beatle song.
Amadeus: "On my '78 Yamaha jumbo, the feedback sound is exactly the same and the tonal quality is the same."
My brothers and I get together every Christmas with our instruments and play Beatle music. One year one of them brought his Gibson J-160e like Lennon's. He set it down near an amp, and, I am not making this up, there was a sound identical to the beginning of "I Feel Fine." Didn't do it on purpose. Could not have done it if he had intentionally tried to re-create the sound.