Final Design:

In order to prevent the motion sensor from perpetually taking readings, I incorporated an on/off sliding switch pictured below. This would allow the user to control when the device was recording motion and illuminating the LED. The switch would be in the "off" position at night, and when the back door was closed. When the door was opened up, and the cat was allowed to go outside, the switch would be turned to the "on" position and the motion sensor would start taking readings and pushing information to the cloud. This solution met the needs of my parents.