AS350B3e Starts when Bat turned on  

OK it was solved after being stuck in field for 9 hours.  Here is what happened and the resolution:

After about 2 hours of uneventful flight time (with 9 normal starts/shutdowns), I landed at remote camp about 60 miles from main base.  Went to idle, and after 30 seconds, I turned the engine swith off.  Approximately 5-10 seconds later, I got loud static in my helmet (radios off already at this point)...after another 10 seconds or so, all screens started flickering and then dimmed slowly as if the battery was dying.  The Garmin 500 screens were particularly interesting...they produced colors I've not seen before...very dark purple shades...almost akin to what I've seen some TVs or phone screens do when the screens have been smashed and the backlight fails. 

Anyway, at this point, all screens were blank and the panel showed no signs of life, despite the master switch being on.  I turned the switch off...waited a few seconds and then back on.  A few caution lights illuminated, and one of the Garmin screens flickered bright white briefly...and after about 2 seconds, the caution lights dimmed and died same as before.  I turned the master battery switch off....got out my phone to get video of this happening so I could share with Mx.  This time when I turne the master battery switch back on, the engine immediate started turning.  It repeated this behavior every time the master switch was turned on. 

Mx came out and began troubleshooting.  In the end, it was the "K3 Starter Contactor" shown below.  There are two relays that are easily swappable (K1/2)...but the K3-7 relays are not.  Each of those has 4 soldered wires that had to be dealt with. 

Yes, we tried banging on the relay, and many other things were checked while waiting for the replacement to arrive. 

As soon as the K3 relay was removed and the contacts could be seen...there was obvious signs of arcing and the contactor "bar" (not sure the exact term...but the metal bar that the relay actually moves to make contact in order to close the circuit) was mechanically stuck in the closed position (effectively siezed in the closed position).  Mx finished installing a new K3 relay and that solved the problem. 

Remaining mystery is the strange behavior of the VEMD and other screens immediately after the initial shutdown.  However, once everything was reassembled, the engine started normally and I flew back to base. 

Thanks to all for the suggestions.  And special thanks to the 2 mechs who came out and dealt with it in single digit temps.  For those who asked, the original poster is the base manager here (yes...also a pilot) who was in contact with us in the field via satphone. 

Cheers everybody.