Agree with everything others have said. Two things I'd add, which you might find interesting
- Several people I know at Bain London have received head-hunting emails asking if they were interested in a role at McKinsey - usually relating to a specific practice (e.g. Private Equity). I am not sure, but I assume there would be similar, lateral poaching in the other directions (e.g. Bain from McKinsey).
- "Cultural fit" is, if communicated appropriately, is actually a perfectly fair reason for switching. Especially at Bain, the whole "collegial" culture is really promoted and believed in by most of the employees. The focus of your reason, though, should be on why Bain/BCG, not on why not McKinsey. You should frame everything positively in terms of what B/B can offer you that is better, rather than why you are not enjoying your time at McKinsey. I spoke to the head of recruiting and they told me that every year we actually hire 1-2 people who interned/briefly worked at McKinsey and wanted to change due to cultural fit reasons.