Can I use Magic Keyboard with multiple computers?

I'm looking for advice on how to best use a Magic Keyboard with multiple computers (or, advice for a wired, compact keyboard to replace my Magic Keyboard).

We have four laptops, and one nice workstation w/ a keyboard/mouse/monitor. We take turns using the monitor, me for work and my family for games.

I almost have a nice one-cord setup, where the keyboard and mouse plug into the monitor, which acts like a KVM switch - any laptop that plugs into the USB-C port gets all of the peripherals. But...

The wired Apple keyboard that I have is too wide for the desk. So, I bought a Magic keyboard. It's the right size, but anytime you plug it into a new MacBook it changes the device it's paired to.

I think the keyboard is probably intended for single computer use, and I can't find a way to disable this pairing feature. Any suggestions?