The name on the certificate does not match, and will not match, the hostname or IP address of your Plex installation on your LAN. Therefore, if you're accessing your Plex Web server via SSL, you'll get the SSL error (line through
https://, broken lock, red background, or whatever other indications your browser gives). This is fundamental to HTTPS, and I don't believe it can be bypassed, though this is something that would depend on your browser. It is, however, only cosmetic--your connection is still there and encrypted, and you can communicate as desired.
If you have "secure connections" set to "required", you can only connect to the Plex Web server via SSL, so you'll always have the SSL error. If you set it to "preferred", you'll be able to connect without SSL, and thus you wouldn't see that error.