If you can somehow manage to install software on your work laptop then I highly recommend Synergy. It is by far a better solution than a KVM and you could keep the keyboard and mouse plugged into your PC so no effect on gaming.

If it were me and I was intending to automate this I would probably do the following. Set up each test distro as a VirtualBox image and take a snapshot so I could easily roll back. Then I would write a script for each distro that downloaded the package, installed and launched the app. I would then probably query the window system to make sure the gui showed up, wait a period of time if I had to and take a screenshot.

This can probably all be done as a set of bash scripts.