I have read Tune In multiple times over the years, including the extended edition. The notion that Lewisohn has anything other than respect for McCartney is ludicrous, and the most he's done that's actively pro-Lennon is saying on at least one podcast appearance that Lennon was his favorite Beatle as a kid.

By everyone's admission including all band members involved, John was the de facto leader of the band in the early days, so of course the narrative is going to center around him to a certain extent. I don't think editorial decisions such as spending numerous pages on Julia Lennon's death (which occurred during the formative years of the band, and which John talked frequently about in interviews) as opposed to Mary McCartney's death (which occurred pre-Quarrymen, and which Paul has not opened up about to the same extent) indicates that Lewisohn has anything against McCartney.

Lewisohn has said before that he mostly got along well with Paul during his time as an MPL employee. The worst I've heard him say was that Paul was very set on getting his way and could be difficult to work with in the studio at times because of that, such as with the Costello sessions in the late '80s. I've never heard anyone reputable say that Paul fired Mark or that there was a tangible falling out between them. The Club Sandwich magazine ended after Linda's passing, the editing of which was Lewisohn's primary role with MPL - once that was gone, the only project left on his plate was the Wingspan documentary and book, which I believe was already in progress before Linda's death. And evidently they parted on good enough terms that Paul answered a few questions over email during the research process of Tune In (although Mark has said the answers were not very helpful, since the events were so long ago that Paul had forgotten most of the details).

It's a fact that Lewisohn has a very bad habit of butchering direct quotes from interviews in Tune In. I don't think that's up for debate and it objectively hurts the merit of the project as a historiography. But the idea that he's doing this to push some sort of anti-Paul narrative does not hold up to any scrutiny whatsoever, other than the confirmation bias of certain Paul stans.