I too have 3 4k monitors at home (in the "office", which is really the shed at the bottom of the garden); all of them are linked to a work machine and my own Ma

I too have 3 4k monitors at home (in the "office", which is really the shed at the bottom of the garden); all of them are linked to a work machine and my own Mac Studio. It's not one button, it's 3, one per KVM, but yep - I just tap the button on each KVM (under the monitors) and the desktop changes, the mouse/keyboard work, and the other machine is up and running in front of me. Even the webcam mounted to the monitor switches over - because it's not just mouse/keyboard, there's a couple of USB ports too. One thing I'd recommend (because otherwise wiring gets unmanageable) is to bunch together the mouse, keyboard, video cables using some split-sheathing. I have an 8-way KVM for the main monitor (all ports used) and this makes it much easier to figure out what's going where.