What is the address range on the two segments?
How are things configured? Can you route between the networks without traversing the Internet?
If supported by your Plex client, you can try directly entering the IP address of the Plex server. Not all clients support this capability.
Depending on how your routing is configured, you may have to give the client a static route to the PMS network or set its default gateway to be the router to the other network.
Plex Media Player for Windows & Mac supports setting manual servers.
- Change PMP to TV Mode (click on the box w/ “TV” inside it).
- Click on your avatar/username & choose Settings -> Manual Servers
- Add the IP address of your PMS server.