Yes, direct experienced-hire moves from Big 4 to MBB happen. The MBA is one route, but it is not a universal requirement. Whether the direct path is realistic depends heavily on your current role, the strength of your project experience, and whether the target office is actively hiring experienced candidates.
Your three years of experience can be enough to create a credible profile if you have meaningful client exposure, strong performance, and evidence that you have taken ownership beyond execution. Strategy-oriented work tends to translate more obviously, but candidates from broader advisory backgrounds can still be competitive if they tell the story well.
The main disadvantage versus MBA recruiting is predictability. Experienced hiring can be more opportunistic and office-dependent, while MBA recruiting has clearer cycles and larger cohorts. That means networking and timing matter more on the direct route.
I would try the experienced-hire path before assuming you need an MBA, especially if you do not otherwise want the degree. If you get little traction after building the profile and networking seriously, the MBA can remain an option rather than being your first move.