I've replied to your email, but to reply here to your specific question:

  > immerse it in water, in a bathtub ready
  > to overflow, and start manipulating the
  > pieces.

  > At what point does the water level rise?

A lovely question. The answer is that the "water" isn't "water", it has to be the same infinitely fine "mush" that the ball is made of. As a result, as you move the pieces out so the "water" ends up forming non-measurable holes to fit them into, and the complement is also non-measurable. So in the same way as the balls kind of "fold out" to become two balls, so the water kind of "folds in" and takes up less space than it used to, exactly balancing the actions of the pieces.