Talk to Pat @harmonyards about your Pittman, he's the expert. FWIW, they are indeed repairable, they come apart quite easily. I've opened a few of them up, but not like Pat who changes the bearings, etc.
FWIW, without hearing the click and the frequency, it's very difficult to hazard a guess as to the source. However, I have my doubts that a "click" is coming from the Pittman, even at slow speed it would be more of a "buzz" I'd guess.
Have you tried the old mechanic's trick? A long screwdriver, touch the end of the blade to the hub of the motor (not the rotating part), and place the other end pressed against the front of your ear opening, right on that flap of skin next to the eardrum. That will amplify the sound and you move the tip of the screwdriver around to see where the click is the loudest.
Removing the motor and running it without the dogbone will indicate if the motor is actually making the noise or maybe that's just what it appears.