Yes you are right. Am using Logitech MX Anywhere 3. Have also spoken to Logitech Support and this is their response:
Here are the steps that you can follow to turn off the Bluetooth mice wake up :
- On your Mac, launch System Preferences.
- Now click on Bluetooth.
- See that Advanced button on the bottom right corner
- Unselect the 'Allow Bluetooth device to wake this computer' option.
If this option is not there, then I would request you to speak with the Apple support as they will be able to help you with this.
Have spoken to apple support as well. They did confirm that apple have removed it from Monterey with no apparent reason, and apple support provided me a link so I can send a feedback to have the advanced button show again. Right now, my only solution is to turn off the mouse everytime i want the mac to sleep. Otherwise the bluetooth mouse will wake it.
Another solution i dont want is to use the included dongle, but the reason i bought the mice is to save 1 usb c slot and just use bluetooth.
Looks to be a widespread issue with monterey 12.1, i wish they would just return it.