I couldn't seem to find this information anywhere, and I just spend longer than I'd like to admit sorting this issue out. I use the Logitech Media Server (v7.9.1 as of now) and I just reinstalled it on my server. After doing that the "Server IP Address" was getting picked up as my public ip address on my gateway server, and not the local network address of 192.168.0.1. This was causing the settings interface to be inaccessible on the local network with an error in the server.log file of Access to settings pages is restricted to the local network or localhost.
Fixing this was as simple as editing the /etc/default/logitechmediaserver file and setting the following line:
SLIMOPTIONS="--playeraddr 192.168.0.1"
Not the httpaddr or cliaddr options which is what I was setting, because I didn't read the help page closely enough.