Well, the car is all back together again. I really can't say it's any better than before; however, I've learned a tremendous amount in the process. After balancing the flywheel twice, adding rubber grommets, replacing the transmission mounts and putting in a new pilot bearing in, I still can't get the vibrations to go away or diminish much.
Given that the second transmission didn't vibrate much at all, even over 7000 RPM, I'm really suspecting the existing transmission has a bent shaft in it or something. There's definitely 10 mils of runout on the input shaft, but I didn't think that would be a big deal. If I look back over my transmission rebuild at the link here, I remember that the bearings in the transmission were shattered. I never really understood why and just replaced the bearings. Perhaps one or both of the shafts were bent already and caused the bearings to break. I'm also guessing that running the transmission through its natural resonant frequency several times over the past week wasn't good for it either. I might be able to salvage a straighter shaft from the second transmission, but I don't know if they are different between the side-shifter and the tail shifter version. The next step is to drive around a bit more and see what happens. Perhaps the transmission will fail, but who knows.
Since the RPM gauge and indicator lights are working, I'm thinking about driving the fuel gauge with an opto-isolated circuit from the 144V pack. It needs opto-isolation so that there are no connections between the 144V system and the 12V system. I'll post that circuit if and when I get it working. Best wishes to all the other EV folks working hard out there to get their cars running.