Using USB-C Switch to connect two computers to Dock

I have a personal and a (new) business computer. I want to be able to switch between them quickly. I’ve been successfully using one computer with three external monitors for a year or more. When I got the new computer, I plugged the USB-C cable from the dock into it and it all worked. I hope to be able to not unplug/plug every time that I need to switch between computers (several times per day). I just got a “USB-C Bi-Direction Switch DATA-VIDEO-PD100W”. The behavior is the same with either computer through the switch: Laptop monitor on, two monitors come on, third (normally replicates the laptop display) remains blank. The display that does not come on doesn’t show up in the Setup:Display monitor map. Any way I can achieve my goal?