Ah I think I get it now! Like this?

My concern with this is that the switch you linked has USBA ports for input and two USB C ports to switch between. The two that are switched between go to the PC and laptop, so how do you connect the docking station and USB switch? I might just have a lack of knowledge, but I'd expect the USB A port is not going to let your laptop/PC send video data to your docking station for the monitor. The USB A connections are normally for an input, like a mouse or keyboard, where as you typically see USB C for something like connecting a device to a docking station.
Maybe you've confirmed this is possible but it doesn't sound quite right to me, though I'm no expert.