@Charles218 So it turns out that 'Works on keyboards with same type' is the correct way to do this for keyboards; though it won't work correctly when Karabiner is being used and takes 'exclusive control' of the device.
From a quick skim, I don't think there is currently equivalent functionality for 'only re-map this on the mouse I recorded it on' like there is for the keyboard shortcuts. I'll mention it in that thread and maybe they can add a new feature for it!