hi Max,
Have you tried " use Service" and see if that helps. I have PowerToys and have this setting so can't say about Microsoft Garage Mouse without Borders.
There are other 3 party apps that are similar to Mouse without boarders
EDIT: I have tried your case and there is no easy way around it, I even tried remote desktop which you only connect one at a time, Window Server might be a better option, it all depends how you want to control a group of computers.