Peter Jackson has said that Paul McCartney told him that John Lennon was the driving force behind the idea.
Peter Jackson eventually brought the trilogy to the screen, starting
in 2001, with the blessing of the Tolkien estate. Jackson recalls
meeting Paul McCartney at the premiere and asking him about the
aborted Beatles film project. Paul recalled “it was something John was
driving and J R R. Tolkien still had the film rights at that stage but
he didn’t like the idea of the Beatles doing it. So he killed it,”
Paul also graciously conceded “it was a good job we never made ours
because then you wouldn’t have made yours and it was great to see
Stanley Kubrick's righthand man Leon Vitali also singled out Lennon when asked about the project in 2018, saying he was "crazy about" Lord of the Rings.
“That was true. That was true,” was Vitali’s emphatic response. “They
came to Stanley’s office to talk about it. I don’t think it was quite
in Stanley Kubrick’s ball-park that idea. Yes, it was something that
they came up with.”
“But it didn’t get very, very far at all. It was just an exchange of
information and people were interested in doing it. But Stanley
wasn’t. Let’s put it that way.
“The person behind that was Lennon. John Lennon. He was crazy about
that story and he was nuts about ‘2001.’ He said that he’d watch
‘2001’ sometimes once a week. He was so fascinated by it. I can see
why he would have connected the dots.”
“I am sure there are many, many ways that you can approach a subject
like ‘Lord Of The Rings.’ Lennon might have thought of it as a
futuristic sort of project. Or thought that the grand scale of ‘2001’
would have been needed to make a film about ‘The Hobbit’ and that sort
of genre. It is interesting.”
So I think the likeliest answer is that Tolkien had a particular dislike for John not because of any personal attributes, because he was the Beatle pushing hardest for a vision of his work that he absolutely disdained.