While $\sf ZF+DC$ does not prove the Division Paradox, it is certainly consistent with it, and in the models that we know of where the Banach–Tarski Paradox is false, the Division Paradox holds.
So, what is the Division Paradox? Your geometric intuition tells you that if you break a stick into five parts, then the amount of "stuff" on the stick is at least $5$. In more mathematical words, if you partition a line into non-empty parts, then the number of parts is less or equal to the number of points on the line.
In the models where all sets are measurable, which are the models where Banach–Tarski fails, we can partition the real line into more parts than points. We can also partition the real line into a completely incomparable number of parts, so there's no comparison between the number of parts and the points on the real line.
Arguably, this is not a geometric intuition. And arguably, this is not an answer since $\sf ZF+DC$ does not outright prove the Division Paradox. But in our current understanding of the universe of sets, one of the two paradoxes hold.
Of course, if you want to be more "grown up" about geometry, you might argue that the Hahn–Banach theorem is intuitive, or at least that if $B$ is a Banach space, then its dual space (of continuous functionals) is nontrivial, i.e. we can separate points using functionals, which is a statement equivalent to the Hahn–Banach theorem. And in that case, you're in trouble. Hahn–Banach is sufficient to prove the Banach–Tarski paradox. So $\sf ZF+DC$ proves that you have to choose between Banach–Tarski or Banach spaces with trivial duals.
(Caveat lector: paradoxes and intuition are subjective and over time we deconstruct and reconstruct our intuition. What is intuitive to one might as well be paradoxical to another.)