I spent 3 years at one of MBB in the U.S. I have several friends who worked at the other two, and we spent plenty of time comparing notes. Here are my thoughts (granted, this is U.S. specific, so you'll have to filter for things that are different in your country).
- McK is MUCH bigger than the other two, which is good and bad. The good part is the network is much larger (both former McK consultants and McK clients), so finding exit options is arguably easier. The other good part is you will have a wider selection of projects to work on. The bad part is the culture could feel less intimate/closely knit.
- Bain has a more fun / younger culture. This is because of their team structure (again, at least in the U.S.). In the U.S., McK and BCG hire 80% from grad programs (MBA, MD, JD, PhD, etc.) and 20% from undergrad. At Bain, it is more 50/50. As a result, a larger percentage of your colleagues at Bain will be recent grads, which makes the environment more lively/fun. Of course, most people who work at McK and BCG enjoy it, so it's not like it is "boring".
- BCG is more intellectual/academic in their problem-solving approach. Bain and McK are super data-driven. You may personally prefer one strategy over the others.
- Bain does way more PE work (McK is scaling back their PE work and BCG is increasing theirs). Doing due dilligence ("DD") projects SUCKS because the hours are horrible. However, you do learn a lot in a very short period of time. So if you're into that type of work, Bain could be a good fit.
- The OP specifically asked about PE exit options. This is only relevant for the analysts (undergraduate hires). MBAs very rarely go work at MBB for two years, then go to PE. The exception is if they go to one of the internal consulting teams at a large PE fund (e.g., KKR Capstone or Bain Capital Portfolio Group). The PE funds mostly hire MBB and I-Banking analysts into their "pre-MBA Associate" programs, which are 2-3 years long. Then like 90% of these PE associates go to b-school. That is actually why it is difficult for an MBA who has spent two years at MBB to get into PE: there are a large number of former MBB consultants who did PE prior to going to b-school, so the PE funds just hire them post-MBA instead. Why take a risk on someone who has never done PE if you can easily hire someone who has already proven he or she is a strong PE professional? That is their thinking at least.
My final point is, you can't go wrong, especially since this is for a summer internship. Pick the one where you like the people the best, and give it a shot for the summer. You can always interview at the other two firms if you don't like the culture at the place you choose for the summer. Good luck!