It´s not the most obvious move, but there are much more weird ones for sure.
As you say, you will need some polishing on client-facing skills and presentation-related skills, but nothing that cannot be done pretty quickly if you are devoted.
About your question of whether to get an MBA, my personal opinion is that is waaaaaaaaaaaaay to extreme tbh. First of all, an MBA has huge implications (cost, money, opportunity cost, going through a very competitive process, etc.). You need a much better reason than wanting to get to MBB to do it. Secondly, even if you did, and even if you did in a target school, that does not guarantee you absolutely anything.
Hope it helps!