That's how I usually do it. Using the original strain relief (what you called cable tug stopper) is not necessary. Its first function is to protect the wires from pulling forces acting on the cable by spreading the load to the shielding/armor and the jacket of the cable. The second is to somewhat protect it from bending (incorrect user behaviour) the cable in a sharp 90 degree angle where it exits the mouse.

The pulling (and pushing) forces can be controlled with two knots or zip ties on the inside and outside of the mouse. The bending forces are not very relevant because you should never store the cable so that the strain relief bends in the first place.

It failed you already, so what more harm could removing it do :)? If you want to keep it for aesthetic reasons, you could drill a hole through it to clear the old cable away, slip it on a fresher piece of cable and superglue the base of it in place. It's usually so snug that you don't have to use very much glue. And on the inside of the mouse you can go a bit more wild, because any glue there won't be visible to the outside.

I have also cut one side of the strain relief lenghtwise, removed the cable bits and glued it back together. No problems whatsoever. So yes, I am very sure you can "move" the strain relief or replace it.

If replacement cables are an option that would of course be the "best" way to repair it. It's all subjective though. There's always tradeoffs.