I have a strong sense of this whole thing--the conflict between Paul and the apostles did not, I think, have very much to do with the Jewish/Gentile issue, as is interpreted in this account. It had to do with Paul--how can I say it--let's say it bluntly--being a false teacher. A false teacher borrows from genuine teachers, claims authority he doesn't really have, is not above bare-faced lies, and mixes and matches genuine teachings with bombastic nonsense--but he looks very spiritual, and he fools people. A big clue is Paul's accusing Peter of hypocrisy. A hypocrite typically projects his failings onto others, especially if challenged--but he doesn't have respect. To me, it's a classic sociopathic profile. I think the apostles knew it--the rank and file were divided--but nobody knew how to stop him.
Where Christianity is at odds with reincarnation, it's primarily Paul's teachings. Where thinking people shy away from Christianity because it sounds nonsensical (like resurrection, as in the bumper sticker which says "in case of Rapture, this car will be unmanned"), or because it seems judgmental, or because of the monstrous doctrine of "eternal hell," that's primarily Paul's teachings. In short, in my opinion, this fellow may have spread Jesus' name to the Gentiles, but he did incalculable damage to Jesus' original teachings in the process. When people accepted his teachings as being identical with Christianity, they left us a religion that is a watered-down shadow of what it could be.
The only way this controversy, of reincarnation being proven while the Bible denies it, is going to be resolved, is to recognize that it is Paul who denies it, not Jesus or the apostles; that in the historical controversy over Paul, the wrong side won out; and that the popular Christianity we know is the legacy of that unfortunate victory.
I would go so far as to say that all the people who rejected traditional Christianity because of the doctrine of eternal hell were the real Christians (whatever label they ended up going by); and all those who eagerly embraced it were the ones Jesus spoke of as regards "gnashing of teeth." The ones who have kind of nervously accepted it because it comes with the whole package, but never really felt comfortable with it, were in-between.