You could confirm if Knoppix supports each network interface card by simply swapping which one is connected to the router and see if you can still ping the router and get out to the Internet (forget about Slug for this test).

Once you know both cards are supported, then it's time to try to get Slug working. Linux should let you bridge the two NICs, but I have no idea how to set that up. Can we assume that you don't have any empty ports left on the router to just plug the cable from Slug in there? And that you would rather resolve this issue and make all Slug traffic go through the Linux box than just add a network switch to the router?