Question: My bluetooth mouse batteries died, and when I replaced them, the mouse didn't reconnect. Now how can I get OS X to discover and connect my mouse without having a mouse to open the mouse dialogue?

Answer: 
You can open the Mouse System Preferences by using Spotlight search. 

1. Press `⌘-Space` to open Spotlight. Type "mouse" to find the Mouse System Preferences and press `Enter`.
2. 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.
I can usually get my mouse to connect if I can get to the bluetooth preferences.

Since upgrading (now on Monterey 12.3.1) that has become more difficult and a lot of the keyboard combinations to get to the menus do not seem to work for me any more.

The ⌘+Space "Bluetooth" to get to the preferences was not working until I found that "System Preferences" has its own option under Spotlight Search Results and it was not enabled. Enabling that has made the preferences show up in Spotlight so I should be able to nudge Bluetooth to connect the mouse next time.