C3422WE, KVM, two PC, Ethernet

I hooked my new C3422WE to two computers, had the KVM configured with no problems.

I have an issue with network connectivity. It seems like the built-in Ethernet connection works only for the active computer? The problem is that my work computer connects via VPN and whenever I switch to my home computer, it loses its connection and I have to reconnect to the office VPN. 

I know that a solution for this is to connect the laptop to a dedicated ethernet connection, but this is cumbersome. Is there some configuration that slipped my search that enables the network connection to keep working even when KVM is connected to second computer?