I known that is possible and i've seen some projects that could be the base for this build (https://youtu.be/0awjPUkBXOU and Tricking Duck Hunt to See A Modern LCD TV as CRT | Hackaday).
My idea was to use the same principle that was described in hackaday project but instead of ataching a wiimote to the gun and make it wierd and heavy, use the wiimote as the sensor to capture reflected IR light like the tracking finger (youtube link) project or emit an IR light from the gun for the wiimote to track.
Making it a little more polished, there could be even an infrared camera atached to the Pi to track the light without the wiimote, making it a standalone product with no wiimote dependency.
The final build will contain the processing unit conected to the video output of the NES, the IR camera with a mount for the TV and a dongle/mod to emit IR light to the camera and flash a light to the lightgun lens.
Both of the projects have open source Code to download, but I believe the only thing that we need is the code for the Pi and Arduino.
I wish I had the proper material to do it, but in Brazil is very hard to get and very expensive. I do own a NES and a Wii but no Arduino or Raspberry Pi.
Hope to see Ben doing it. I think everyone is going to love to see that, specially because Ben can make it look like an actual product and not just a prototype like the links I shared.