CD changer spindle

[Bald Eagle]

Without going into the gory details about how, I believe that the shaft for the drive spindle on my Sony CDP-CX230 CD changer is bent or at least experiencing excessive runout.

With the case open, you can watch the disc shimmy back and forth perpendicular to the motor shaft, and this gives rise to an annoying click 1-2 times the frequency of the RPM.  I can gently stick my fingertip into the center of the clamper and with faint pressure completely stop the clicking even though I can still feel a little wobble.

Are there simple effective ways I can fix this problem without having to try to straighten the shaft by bending?  Can I weight the spindle or clamper like balancing a tire?