The old MKIII platter is "pressed" onto the spindle. Once you break it free using the method above you will neet to reseat it if you want to use it again. Use your 2 piece clamp to do this. It should give you enough torque to pull the spindle up and seat the black platter firmly again.
Check to make sure all is level though once you do this. There could be a little wobble in the platter (it is a machined fitting so not perfect). If there is then just unset them again, turn the platter 1/4 turn and reseat it. Keep doing this until all is level and smooth.
Enjoy,
Bob