the way a light gun works is going to be inhibited by any kind of digital processing you may introduce.
the way it works:
the light gun has a sensor that produces a pulse when the light entering it reaches a predetermined threshold.
when you are playing the game, the sensor board outputs nothing. when you pull the trigger, 2 things happen... the computer starts counting...and the computer sends a frame of white to the monitor.
monitor begins overwriting the image with white...the time it takes for the beam to scan, starting from the top left...until the sensor picks up the light from the screen turning white is where the position is. this is usually done ~10-12 times to get an average of position to account for noise or voltage fluctuations. (at least on the big buck hunter games it is)
the time for a single frame to be "written" to the screen is 16.67 milliseconds.
can you see where the problem is?
any kind of convertor is going to introduce frame processing time and introduce lag to the image. While this wouldn't be a problem with say a fighting game or space game where a delay of 10 milliseconds to display a frame is no big deal... but 10 milliseconds in the scope of 16.67 milliseconds...the computer is expecting the screen to have been 3/4 drawn by now.
thus you end up with a gun whose aim is nackered in the corner no matter where you shoot.
that said, lots of CRT monitors support the resolutions these gun games run in. If you can manage to run it in it's native resolution, it will work no problem. most Sammy games (extreme hunter etc) that run on the Atomiswave platform will run at 640x480 with a couple flips of some dip switches...and they also have a VGA output on them which is perfect for just plugging in a VGA monitor.