Howdy - I did exactly that to join BCG, and helped someone else do that as well although it didn't work out for him, at least not right away... but he ended up at McK, not bad either :).
First off, the 3 firms have different views of this process. In my experience, Bain is very unlikely to take an experienced hire while McK is the most experienced hire-friendly. Some clients will forbid MBB to poach their staff, but I think most will let you apply if that's what you want.
In my own case, my company was a regular BCG client, and I ended up working alongside BCG on a 4-month case. Near the end of the engagement, I approached the principal and told him I wanted to apply. After some discussion re. my educational background, interest and level of commitment, he agreed to support my candidacy. To my knowledge, he never specifically asked my employer if that'd be ok since my VP only learned about it when I resigned. I still had to go through the normal application / interview process, but having a current consultant support my application at least helped me get an interview. There's never any guarantee, and the longer that consultant has been in the company the better, but if you get a principal or a partner to put their name on your application, you are in good shape.
In decreasing order of value, I'd place "recommendation from a partner" first obviously, though it will be hard to get; second, "recommendation from a consultant who's worked with you", then "knowing the recruitment team". Getting a recommendation from a low-level consultant who has NOT worked with you has nearly no value, and I'd question said consultant's decision to recommend you if he/she doesn't know how you work anyway -> a consultant needs to build his/her credibility to move up, recommending a quasi-stranger is a high risk proposition.
Looks like you work w/ MBB teams. Your best bet is to impress the Principal or Associate Partner and approach them just before the end of a case. The partners probably won't spend enough time with you to know (and would ask the consulting staff for feedback anyway), and the actual consultants/associates do not have enough pull internally.
PS: if you decide to go this route, make sure to start preparing for the case interview sooner rather than later, things can go extremely fast.
Does this help? Let me know!