I'm keeping everything small here to start.  Actually, I think that may be a requirement.

The filament loops will want to deflect as I apply torque to the wheel.  So the size of the loop needs to be just right to resist that deflection, but then still allow the loop to provide some spring tension and act as a tire.

A larger rim diameter would allow me to place more loops and spread out the off-axis force for the 'against the grain' Mecanum motion, but will also increase the force required to start the sliding motion.  I totally guessed at an appropriate balance here, and so the loops are elliptical with a short radius of rim_r/1.44 and a long radius of rim_r/1.44*1.25.  This balances out to a roughly spherical tire cross section though due to the angle of the loops.

The finished tire I've modeled here including the loops is about 90mm in diameter and has 30 filament loops.  The filament enters the rim at a 115 degree angle, so I more or less have 3/4 of a tire circumference with the filament strands.

This was all based off some nylon line I have for my string trimmer here which has a 2.8mm diameter, but it is octagonal instead of round.  The ridges on the line should enhance the grip in the off-axis direction, but still allow sliding along the length.

Or so goes the plan...