Bluetooth device not showing in Sound -> Output

Try and boot the Mac in the safe mode to see if you can find the speaker in the safe mode. If you can, there is a problem with external software when not in the safe mode

Steps for booting your Mac in the safe mode differ wether you're using a Mac with apple silicon or an intel mac

Start up your Mac with Apple silicon in safe mode

1. On your Mac, choose Apple menu  > Shut Down.
2. Wait for your Mac to shut down completely. A Mac is completely shut down when the screen is black and any lights (including in the Touch Bar) are off.
3. Press and hold the power button on your Mac until “Loading startup options” appears.
4. Select a volume.
5. Press and hold the Shift key, then click Continue in Safe Mode.
6. The computer restarts automatically. When the login window appears, you should see “Safe Boot” in the menu bar.

Start up your Intel-based Mac in safe mode

1. Turn on or restart your Mac, then immediately press and hold the Shift key until you see the login window.
2. Log in to your Mac.
3. You might be asked to log in again.
4. On either the first or second login window, you should see “Safe Boot” in the menu bar.

Try and reboot the Mac and the speaker to see if the issues persist