I adjusted the cable tension plate mounting bolts so that the plate doesn't rub against the side of the rotating spool. And I also installed a UHMW Polyethylene plastic sheet between the plate and the cable.
But unfortunately, that didn't reduce the noise. It seems like my original assumption was incorrect: the noise isn't caused by something rubbing.
The plastic sheet I installed actually caused a new problem: it didn't hold the cable in place properly so the cable walked off of the spool and got jammed between the spool and the motor housing. So I removed the plastic sheet.
It occurred to me that the truck rack might be vibrating -- causing the excess noise. The rack doesn't have any loose parts that are rattling. But I suspect it's still resonating from the winch vibration.
So I isolated the winch from the truck rack by padding the winch with rubber pads.
I cut a piece out of a mud flap to pad the U-bolt.
And I used a thick rubber pad (possibly made from old tires) to pad the winch.
That worked. The winch motor is still somewhat loud. But the rack isn't making a jarring vibration sound anymore. So the sound is tolerable; I no longer need hearing protection when operating the winch.