Just an update.
After having the Le Mans apart for a couple of weeks I finally got it back together today. I installed the all new GEI relays that were recommended to me and fired her up. No immediate problems.
I've started it about 10 times with no glitches. I rode it around our bay (20 miles), no problems. Other problems that I have been experiencing (stutter in one cylinder @ constant 4K rpm, sudden electrical loss draining battery leaving me high and dry), gone (at least for now).
I'm hoping this one, simple fix may have taken care of all the electronic gremlins I've been experiencing over the last year.
On my "concerns" list is that I replaced all my relays less than 2 years ago with the same GEI relays and that one (or more) of them has failed. They're not supposed to do that. I'm still somewhat concerned about the reliability of my bike but will happily proceed with my testing (read as "riding").