How can I connect a bluetooth mouse without using the cursor (chicken and egg problem)
You can open the Mouse System Preferences by using Spotlight search. 


Press ⌘-Space to open Spotlight. Type "mouse" to find the Mouse System Preferences and press Enter.
When the Mouse Preferences are opened, make sure the mouse is on and wait for it to be discovered. The "Connect" button should be blue, so you can press Enter to connect to it, and you're done.