Connecting Bluetooth mouse with keyboard only

I have a Mac mini and the Bluetooth mouse is rarely recognized on boot. I use Alfred to get me to mouse settings and see the devices, but can’t tab over there with keyboard. I also have mouse setup in tooth fairy…is there something I could do there with keyboard navigation? (same behavior with Magic Mouse or Logitech mouse)

You could set a hotkey for the mouse in ToothFairy’s settings:

and then connect the device just by pressing the key (or key combination), without having to navigate anywhere.