If you want to place the cases with your right hand, you might try turning the shell holder so that you can insert and remove the cases from behind the post where the primer device mounts.
If you want to place the cases with your left hand, change the angle of the operating arm to provide more room for your hand. Higher will let you insert from the front and lower to let you insert from the left (getting behind the operating arm).
Of course, the ultimate fix would be to replace the operating arm with a very short arm operated by foot-operated system.
Sorry about your wrist. I work with a computer all day, too and have problems with sore wrists. I gave up on using a mouse entirely and use a trackball. It is much easier on me. I also use two trackballs, one left of the keyboard and one right of the keyboard. I can switch off when I want and give relief to whichever wrist is getting tired. Kensington and Logitech make nice, symmetrical trackballs.