Banach-Tarski describes a way of breaking apart an object into several pieces, and re-assembling them in a way which does not preserve mass in this simple mass-preservation model. Thus, a process analogous to this is not possible in this matter preservation model.
The immediate consequence is that the matter preservation model I have described is not a viable mathematical model of physics in which some sort of "Banach-Tarski process" is possible; and equivalently, in any world which is described by matter preservation, there cannot be any process resembling what is described by the Banach-Tarski theorem.
Could it be that there is a matter-preservation model in which Banach-Tarski is possible -- for instance, if point masses are possible? Maybe; but then what's the point of talking about tearing apart the empty space in between into unmeasurable sets and re-assembling them? You take apart mostly empty space, and re-assemble it into mostly empty space; whoop-de-doo.
So, with respect to simple models of matter conservation, Banach-Tarski seems at best peurile and utterly uninteresting; and at worst, utterly unrealistic because a re-assembly process analogous to the Banach-Tarski would not perserve mass.
Now, matter is not actually conserved in our world, but matter with energy seems to be conserved. Perhaps tearing apart a sphere into immeasurable pieces requires a large amount of energy: but we don't have a good model for this, and making models of the world for the sake of theorems instead of observed data is not good science. It is in principle an interesting way of building world-models, but it is not clear that this is how science should work.
This is the basis of my argument in the other thread. I was working with an intuitive model of matter preservation, which can be formalized into what I have described above.