If one is hard mounted, no need to deactivate it moving the cursor, or you could put a piece of opaque tape over the sensor.
However, I am wondering why you don’t just do everything with your left hand. It will seem awkward at first, especially with it being your non-dominant hand, but you will build up more fine muscle control and get much better at it over time.
I had to do that long ago, being left handed, but so often encountering workstations with the mouse on the right side, or the mouse wasn’t ambiedexterous.
I should also mention that (on systems I’ve set up for my own use) I do my mousing with my wrist planted still, all mouse movements are done with fingers. This works well with a good laser mouse, not so much on a battery/power conserving LED sensor cordless mouse, as they often have too much lag and too low a sensitivity for good tracking of smaller movements.
Once I get finger arthritis I may change that, start using my arm to move the mouse.