From today's perspective, it seems glaringly obvious that there was no way not just John but Ringo and George would have accepted Paul's in-laws as managers. However, I really think a lot of that estimation is hindsight. Ringo himself said that if "Lee Eastman had been Lee Northman", he'd sided with Paul. It was by no means a foregone conclusion in his case, it wasn't in John's, see above, and I don't think Paul was aware of the amount of George's resentment of his bossiness in 1969 (for evidence, see not just hindsight, but Let It Be conversations about George during his temporary walkout). And the core of the matter was that they really, direly, and quickly needed both someone capable of being a fixer due to the Apple mess, and capable of managing them. Which is why I don't think Paul would have gambled at this point - he was far too aware of how much of a mess they were in. He truly saw the Eastmans as the solution, the only solution.

(nemperor once listed the possible managers for the Beatles as of 1969, and it turned out there really weren't that many options if you go for a) successfull track record and ability to play in the big leagues, and b) availability. Now one interesting question to me is why Paul didn't consider Allen Klein, who was one of the few options, before Klein met John. He knew about Klein and had brought up the famous Stones record deal to Brian Epstein, plus Klein had made no secret of the fact he was interested in getting the Beatles even during Brian's life time. I know Mick Jagger later was less than complimentary about Allen Klein, but that was AFTER John had already introduced him as his choice for manager to the Beatles, so "bad word of mouth" probably isn't the explanation, and "intense dislike" was only to develop during the course of 1969.)

All this being said: I wouldn't be surprised if Paul, on general "John vents what I can't allow myself to feel" principle, would have expected some John-Lee clashes in their further future with a certain degree of anticipation. But that would have been a Klein-less future in which E & E was securely the new management. IMO he definitely didn't expect - and nor did Lee Eastman - the initial John L - John E encounter to go as badly as it did so that Lee had to show up himself in London.

More about the song(s) question(s) tonight, must be off to several appointments.