From your description of the problem, it seems that Couatl was still running normally but, something in the sim caused it to lose communication with the simulator.
Try to turn on logging in the Troubleshooting section of the GSX Settings page, and post a Couatl.LOG file AND a Couatl.ERR file (if there is one), after it happens again.
I restarted P3D and left the aircraft at the gate at KLAX again and waited. This time it ran for well over an hour. I was panning around and checking different things around the airport for a few minutes and then left it alone again. After 2 minutes I watched the simulator go into pause and all the terminals disappear. I un-paused the sim and checked the Add On's menu and again Couatl was missing. Add on Manager was still present.
This is not very clear. If it ran for over 1 hour without issue, it's clear there aren't any problems with normal usage.
What is not clear is when you say that, after you "left if alone again", it went into Pause mode. What does it mean ? Is your sim going into pause mode automatically after 2 minutes ? Or, are you referring to your PC power saving Standby mode ?