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: 
If the batteries run out in my wireless mouse it never reconnects automatically and I have resorted to reconnecting a wired mouse to click on the bluetooth icon in the menu bar. Then I found that you can activate the bluetooth icon by pressing Control-F8 (the "Move focus to status menus" in the keyboard shortcuts) and using the cursor keys and the enter key to get to the connect menu item for the mouse.
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.