Jim,
I have the same problem you have and for me it was the tires.
The tires, although you re-balanced them, maybe improperly worn - meaning you may have a slight alignment problem. If this is the case, you could not detect it with the naked eye just by looking at the tires. It might be slightly off than ideal and this could cause vibration problems at certain speeds (due to harmonic vibrations).
I know you might not have spare tires available, but, as a test, I would replace all 4 wheels temporarily and drive it and see if there's a difference. This could save you the costly expense of replacing motor mounts, transmission mounts, driveline axles, etc. only to find out that it was just the tires...
If you happen to know of another friend with an LS400 year 90-92, and he/she would allow you to swap tires, then this might be a cost effective test to troubleshoot.
It would be a shame if spent money on parts that did not need to be replaced.
Just my 2 cents... Hope you can find out the root of the problem...Good luck....
Teamer