Which MBB is best for PE Recruiting?

I'm going to frame my answer based on the implicit assumption you really want to do PE and want the best shot at it: You're asking the wrong question

Why? Because recruiters from search funds and PE firms typically recruit associates out of IB and CO by tiered buckets (i.e. MBB, T2, etc). The anecdotal cases you hear of people moving into PE are colored by individual connections of individual people, rather than the overall institutional relationships.

Here's an example: One of the Partners on my study went from our MBB to a MF and back (he was not a fan of the culture; also keep that in mind if you're so dead set on PE). This firm doesn't technically recruit directly from us, but I'd have a leg up if I decided I wanted to work there because I have a working relationship with him, and he left on good terms. The "Bain is 1A, McKinsey is 1B and BCG is 2" that you hear is an echo chamber from people reading too much into these one off occurrences. Bain focuses a higher proportion of its work in DDs, as such, the typical exit is more prone towards PE. Ergo, the Bainies you speak to are more likely to know someone who left for X fund, not because having a red arrow in a circle on your jacket magically grants you PE mastery.

Here's some food for thought: Statistically, McKinsey does the most DD's of any consulting firm and its not even close. So now does that mean McKinsey is the top dog for PE? Maybe, maybe not: you could very well make the argument that this makes it too easy for McKinsey analysts to compete with each other for PE spots because they all have easy access to DD cases. Regardless, this emphasizes my point here further: all these "stories" and "stats" don't really do much compared to having established relationships with key figures in each firms' respective PE practice.

TLDR: 80/20 it silly; the M vs B vs B for PE brand debate is the wrong 20% to focus on