Logitech Solution ... in the hopes of helping someone else learn from my mistakes, here is the secret to disassembling the Marble Mouse from Logitech...

There is a hidden screw! The bottom of the mouse with the model, serial number and other information isn't really a solid piece of plastic, but a stick-on label made of material exactly the same color as the plastic of the base. If you peel away that label, you will see a deep shaft with a Phillips head screw running right up through the center of the mouse. Undo that and you can lift the top of the mouse off for cleaning.

I, unfortunately, didn't discover this until I ripped my mouse apart by brute force.