I called Line 6 today. After about 40 minutes in the phone queue, I spoke with a support rep who was at first skeptical of the idea that a customer could service a day-old JTV-59. After much discussion, he put me on hold for about 10 minutes, and when he came back, he told me that he'd spoken with a repair tech and ....
.... what I needed to do was to remove the encoder cavity cover, then unscrew the two small silver screws on either side of the model encoder bracket. Next, remove the model knob and gently push the encoder shaft to pop the assembly out of the guitar. Invert the encoder assembly and remove the two small black screws which fasten the LED circuit board to the assembly, then remove the LED board itself. Don't disconnect any wires. With the encoder assembly hanging loose, and the LED circuit board detached from the assembly, the shaft nut can be tightened with a 12mm wrench. Add a little LockTite Blue for security. Reassemble and rock on.
This took me about 15 minutes. Done, and happy.