Interesting.
I have an Everdrive N8 I use for testing, but I don't have an N8-Pro, so I'm not familiar with N8-Pro specific features. I don't suppose there are some settings in the N8-Pro that could be a factor. Would I be correct in presuming that your Zapper is plugged into controller slot 2?
Also, regarding the Zapper not working. Are you getting any response when you pull the trigger? If so, does it flicker the screen to black when their are targets on the screen (either in the 2-choice menus or when enemies are on screen)?
Last aspect to check, what type of device are you playing the Everdrive on? Is it a stock NES or a third-party machine like a Retron? Is the device plugged in directly to a CRT TV or does it go through another device along the way?
Sorry for the volley of questions. These are just what came up as I started thinking about what might be going on.