I'm going to try and help you by disagreeing with a few of comments / assumptions.
Firstly, I don't think that it's difficult to make that transfer. There are more natural entry points to PE for sure, but with the right experience at MBB, you can easily demonstrate the right knowledge and skills. I have former colleagues who spent much of their time at MBB on Due Diligence projects and that was a perfect lead-in to MBB. Also, Bain specifically is certainly perceived as a strong consulting partner for PE firms, so they would certainly hire from Bain.
Second, I'm always surprised at the significance people attach to specific office prestige. For goodness sake, it's MBB! The type of work you did, the knowledge and skills you built, and your networking to "sell" the same will by far supercede the specific office you worked in.
Finally, join an office that gives you the greatest opportunity to do the work you want to do with the people you connect with. That will give you the greatest opportunity to succeed at MBB at build up a PE-resume, regardless of the office. Which, again, is much more important than office prestige.
Hope this helps,