About the Banach & Tarski theorem

In other words, the Banach & Tarski theorem describes a process that implies a physical model which is different (and incompatible) with LCME (a currently accepted description of our world). An instance of a process that is possible and entirely compatible with a physical model implied by the Banach & Tarski theorem could be judged to be similar or analogous to an instance of a process that is compatible with LCME. A process that is analogous to a Banach & Tarski process could indeed be observed in our world, depending on the exact definition of analogy. In order to complete your argument, you would need at least to provide a definition of 'analogy' or 'similarity'. Now, maybe you think you've done that, by presenting the LCME physical model and how it is mathematically inconsistent with a physical model implied by Banach and Tarski. The problem is that of analogy. Your unstated premise is that no 2 instances of a processes described by incompatible physical descriptions can ever be 'analogous' or 'similar'.