I believe what you’re looking for is called a puck light.
This puck light from GE matches most of your criteria
Single lamp—we don’t need a pack of 6
Well, it’s a 2-packAA or AAA batteries—I don’t want something that needs to charge
Takes AAA Batteries
Dimmable or different light settings
Well, it doesn’t have different settingsSmall enough to put in night stand when finished
YesNo sticky back
The 3M tape is separate, so you don’t have to make the back sticky