Assuming the laptop you're looking to control has HDMI out and USB input for Keyboard and mouse, I think you're right with the KVM switch idea, one that supports USB and HDMI input, and can switch between them between two devices. What I would do is get something which can record HDMI on your main PC. Some gamer devices have HDMI passthrough, which you'd plug into the KVM switch, but you could also use an HDMI splitter to have a feed from the laptop going into the KVM switch and to the recorder on your main computer. On your main computer, you could use OBS Studio to record the video from the laptop.