I don't understand what you think the "traditional resolution" might be. In the Banach-Tarski theorem you are partitioning a 3-dimensional solid ball into finitely many pieces. Because the ball has uncountably many points, those pieces will have uncountably many points.[0]

Does that help?

[0] Actually that only shows that at least one of the pieces must have uncountably many points, but in the theorem we find that at least four pieces must have uncountably many points.