Rav Dovid, this is another difficult question!

1) I found, bs'd, in the Birkei Yosef by the Chida (Shulchan Aruch, Choshen Mishpat 12, in Shi'urei Berachah, page 54), that he writes that one cannot say that the reason one says that three are required for Din according to Rebbi Meir is because we compare Pesharah to Din. This is because there is a strong Sevara that two are sufficient for the Kinyan. Therefore, we must say that when we compare Pesharah to Din, this means that if there was a Kinyan, we compare the Pesharah to the kind of Din where the litigants had accepted upon themselves that two judges would be sufficient.

(See Beis Yosef on the Tur, Choshen Mishpat 3:3-4, that it is only where the litigants have not accepted upon themselves the two judges that we say that if two judged that the Din is invalid, but if they accepted upon themselves the two judges, the Din is valid.)

2) However, if Pesharah does not require a Kinyan, this is comparable to a Din where the litigants had not accepted upon themselves the judges. If so, three judges are absolutely essential.

3) In short, if Pesharah needs a Kinyan, there is a very strong Sevara that two suffice. This is a stronger Sevara than the possibilty that we should compare Pesharah to the sort of Din that requires three.

4) I think that I have found an answer to this question in the words of the Yad Ramah on the Sugya.

a) The Yad Ramah (DH v'Diyek Rav Ashi) adds a few words to what Rav Ashi said, that if one would maintain that Pesharah requires a Kinyan, why should one need three? He writes, "Why would one require three to be a Beis Din, where even though no Kinyan was made, one could still not retract?" One senses from this that what is special about a Beis Din is that it cannot retract. If so, we can say that if Pesharah requires a Kinyan, then one cannot retract from Pesharah either, so since two is sufficient for a Kinyan, Pesharah should be equivalent to Beis Din, and two is enough.

b) When I saw the Yad Ramah, I started to think that possibly he learns that what the Gemara means by saying "Makish Pesharah l'Din" is that what is special about a Beis Din is that what the Beis Din rules is binding and cannot be reversed. Now we know that Rebbi Yehoshua ben Korcha says (on 6b), "Mitzvah li'Vetzo'a" -- it is a Mitzvah to do Pesharah, to make a compromise. But the problem about Pesharah is that it may not have the authority of a Beis Din since either party might be capable of backing out if they are unhappy with the Pesharah made. However, Rebbi Meir said that Pesharah must be compared to Din. This does not necessarily mean that both Din and Pesharah require three. Rather, what it means is that both Din and Pesharah must be irreversible processes. If one can make Pesharah irreversible without three, this is also acceptable.

c) Then I looked a bit further in the words of the Yad Ramah. After he cites the Gemara below that the Halachah is that Pesharah requires a Kinyan, the Yad Ramah writes that we should investigate what was the argument between the Tana'im concerning Pesharah. He writes that if they made a Pesharah and did a Kinyan on this, then even Rebbi Meir, who usually requires three, will agree that since everyone wanted to make a Pesharah, this will work even if the person making the Pesharah is a close relative or is disqualified as a Dayan. This is no different to a scenario where they accepted a close relative or a Pasul to be a Dayan and made a Kinyan on this, when the Din would stand and could not be retracted.

d) We see that the crucial factor is that the Din cannot be reversed. If Pesharah cannot be reversed, then it can be compared to Din, and since two are sufficient to make Pesharah irreversible, it follows that we say "Makish Pesharah l'Din" according to Rebbi Meir with only two.

Kol Tuv,

Dovid Bloom