You can check the battery status of your mouse using the Logitech Control Center (LCC) mouse and keyboard software for Mac.
NOTE: If you don't already have LCC installed, you can download it. See answer 360023357933 for instructions.
To check the battery status of your mouse:
Launch the Logitech Control Center.
Apple Menu > System Preferences > Other > Logitech Control Center
In the Logitech Control Center device window, double-click the picture of your mouse.
In the top-left side of the window, click General to view the battery status.