I didn't read all 100 replies.
This service bulletin covers what I believe has already been discussed. If all those tests are OK, the new regulator is about all it could be that I can think of.
I found this by Google "John Deere 322 boiling battery"...
Green Tractor Talk › ...PDF
322 battery boiling - service bulletin from Deere.pdf - Green Tractor Talk
Yes. I agree. I ordered a new key switch and after that comes the relay fix if it doesn't change things. I can see why farmers would butcher the electrical systems on tractors just to keep them running, electrical gremlins are time killers.