Getting into MBB is always hard (there were 60 applicants for 1 position in my office around the time I joined), so while there's obviously some skills, don't forget also that some of it is a numbers game, and some of it is plain dumb.
Coming from boutique, the "why consulting" is already taken care of, and "why McK/BCG/Bain" probably easier - especially if you interact with some of these guys / share a customer. You will hopefully also be used to develop a hypothesis and build issue trees / develop MECE structures.
All in all, I think this is a great first experience which will prepare you well. And as Vlad pointed out, you absolutely should apply to all 3 MBB (and to some higher-rated Tier2 as well, such as ATK, RB or Strategy&)