Hi Chris,
I have found that you must still do something manually, turn "on" only the mouse you want, and when you are done turn "off" the mouse.
This is what I have found: rename the mice (System Preferences). In the Bluetooth pane, delete the mouse in both cases ("home" and "work"), then use Bluetooth to find a mouse, for example, the "home mouse". Do the same for the work mouse, now, they should be more easily identified. At work, turn on only the "work mouse". The same at "home", turn on only the mouse you want.
Apple says that you right-click and choose "rename", but I have to click while holding the Control key, and then I select "rename." In my case I added the year of manufacture to the existing name.
My two computers are iMacs (each with a Magic Mouse). . . I don't know if it will work for your notebook, but this solution works for me.
I just realized that you have two "Magic Mouse 2" mice, which are more recent. My two are 2009 and 2015, battery operated. It may still work.