Remote Listening is VERY Unreliable

Sounds like your router is deleting the automatically created port forwarding after a while.
The solution is to use a static port forwarding which you create manually in your router.
Then you only tell Plex which port number to communicate to outside clients.

Since this is Windows, do also disable any power saving features of your network adapter. Particularly (but not exclusively) those which allow it to reduce the link speed when idle. These often lead to large delays in network communication.