Finally, an update! After the installation of a third steering stabilizer, which improved things a bit for maybe 1,000 miles, the shake appeared again with a vengeance. I thought about it and decided it was time to order a Fox 2.0 (basic, cheaper version, not adjustable) and see how it did, after hearing from a couple of folks that said they had a good result. Well, it took care of the shakes nicely. The steering felt better than ever and the shake when hitting bumps was gone. I know that was treating the symptom, so I then Friday finally installed the Rancho track bar. Now the steering feels even better. At this point, I'm declaring the steering shake cured. If it comes back, I'll look at Metal Cloak adjustable lower control arms to change the castor angle, but I am optimistic that won't be necessary.
Thanks Eddie for all the help, especially the Rancho track bar. The fact it centered up my front axle was just gravy. This thing went from the worst steering I've had on a vehicle (not dangerous, but seriously annoying) to feeling fantastic. It certainly feels better than any Jeep I've driven, which is how everything on the JL was intended to be.
I'll check in here and there as the miles pile up, both on and off the road, to see if the fix sustains itself.