Finally got the packing unit off the plastic coated plunger rod. Fortunately, the plastic coating on the rod is unblemished.
In the following pictures you will see (roughly from left to right): the threaded packing unit, the coated thick rod, bits of petrified felt that was inside the packing unit, a whole un-petrified felt ring from another thick-rod packing unit, the hard backing washer, and below it, the closing nut.
Here's a close up side view of the double threaded packing unit.
Here are some shots of the inside of the emptied packing unit. You can see a very wide opening for the thick coated rod.
This looks into the packing unit end of the barrel. You can see a "floor" made be a piece of plastic that has been swaged (?) into the barrel. The packing unit would screw down tight (counter-clockwise!) onto this floor. What you cannot see are the remnants of the rubber O-ring/washer that lay on the "floor" and formed a tight seal between the floor and the bottom of the packing unit. Judging from these remnants, the original o-ring was the size of the large flexible piston washer with a suitable wide hole in the center for the thick rod.