If it continues, I'll take you up on that.  After repositioning the ferrite bead, the problem (for now anyway) seems to be gone.  The bead is just floating around in the control cavity... is it supposed to be attached to anything?

Well, the pickup leads need to pass through it - otherwise the bead is of no use.

They're running through it, but the bead is not otherwise attached to anything.  I think I understand that the bead is just a non-conductive shield, so I guess it doesn't matter.

Correct - it doesn't need to be attached. You can stick a bit of styrofoam to it to keep it from rattling against the back plate.