I've taken the bottom of the joystick off before when one of my axises (whats the plural of axis?) started messing up. One axis was working perfectly but the other had almost no response or it was jumping around ridiculously. I took the bottom off, quite easy, 4 screws, and took a look. I found the 3 wires from the joystick input to chip, disconnected the wires and reconnected them. Joystick worked fine.
But I wouldn't think you would need to mess with the base at all to take apart the handle. I see 4 screws on the right side of the handle (recessed into the handle). Looking closer at it you'll need an allen wrench to take apart the front "shield" part, again 4 screws. The problem is the small chips, springs, and wires running delicately through the stick. They should stay in place if you take apart the handle carefully. After that you have to get everything back into the right place so you don't jerk out any wires or make other buttons not work.