RetroGaming

If going the emulation route and willing to use a different kind of gun, you just need to feed the XY coordinates to the emulator. Duck Hunt with a Wiimote is pretty trivial to implement. Once the emulator knows the XY coordinates it can just feed the correct inputs to the game no matter the latency of the gun: if the emulated screen is a white pixel at that coordinate, then the controller port gets a one, and circumvents the entire problem. It can probably be done on original hardware too with an RP2040 to intercept the video signal and inject the correct input, as long as it knows the coordinates before sending the fire input to the console.

To make work on the original gun, the emulator could rewind and replay with the correct inputs fast enough to be completely invisible to the user, we already do that for netplay over the Internet. Play once, process the gun input, rewind and replay with the correct inputs (in the background) and then continue at the exact frame we rewound and it's completely invisible except a tiny bit of rubberbanding.

Original console and original gun though, it's tricky but if we could frame quadruple the thing to 240Hz and use an OLED with zero input lag, we could theorethically have it displayed in time for the console to be able to read it with the light gun by vblank. The tolerances of that would be insane.