Magic Mouse 2 won't connect on login screen

When I power on my Mac Mini running macOS 10.15.6, my Magic Mouse 2 often won't pair, which makes it impossible to log-in to my account--I have several user accounts that appear on the log-in screen. Therefore, I have to connect a wired mouse so I can choose a user to be able to enter a password. Once I log-in to my user account, the Magic Mouse 2 connects via Bluetooth.

This situation occurs most often after I've connected the Magic Mouse to my MacBook. It remains impossible to connect, even if I've shut down the MacBook. I also have tried to connect the Magic Mouse via USB to the Mac Mini during startup, but that doesn't seem to work.

This really just seems like a very annoying bug. I will probably just switch to a wired mouse.