"Your CD key is invalid or in use by another player "
This is the error you would encounter after attempting to join in a LAN multiplayer with a 3rd PC.

Below a work around which works for me:
Immediately after the error has been generated go to the computer which created the game and

1) Under GOG Games\Star Wars - Republic Commando\GameData\System
2) Find the SWRepublicCommando.log.txt file and Open it
3) You will be able to identify the two other players joining the game with two autogenerated CD KEYs
The 1st player joining should be authorized and the 2nd player rejected: CD KEY IN USE and then session canceled
4) Copy the CD KEY generated for the 1st player.
5) Close everything
6) Under your PC Start icon (Windows 10) type RegEdit and open the Registry Editor
7) Under the Registry Edit use FIND to search for CD KEY under HKEY_LOCAL_MACHINE\SOFTWARE and find the Republic Commando CDKEY Path:
\HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\LucasArts\Star Wars Republic Commando\1.0
8)CLICK CD Key And under the empty value data paste the CD KEY copied earlier from the log file.
9)This needs to be now the 3rd player to join and now you should be able to play a LAN multiplayer game with 3 players.
10) I have not try more players but should work applying similar steps.

Looks like the more simple resolution to this issue, in the case of the GOG version of the game at least, is for the host to use the "Create Internet Game" option. Then any number of peers may connect with "Connect to IP" without being kicked out due to the cd key error - the error ALWAYS shows up if the host uses the "Create LAN Game" option, for some reason...