You can definitely use the PS2 gun as almost all light guns use the same tech. It's the I/O that differs.
I use a Time Crisis arcade gun with a PS2 by connecting it to a Guncon 2 PCB instead of the arcade I/O. The gun itself is not modded (as it's just a photo diode inside).
The question is how does a 246 with Time Crisis 3 recognize a guncon 2 board. My guess it that it would not recognize it as an I/O board. The PS2 TC3 version has custom Guncon 2 drivers on the disc.
BTW, you can already play emulated versions with camera based guns on HD flat screens. You've been able to for years. It's just that none of them are as good as using a real light gun on a CRT display. Plus, none of the emulators play as well as real hardware for Time Crisis 1, 2, 3, or 4.
Time Crisis 5 was already PC based and since it's been cracked, it's the exception.
The best PC arcade gun solution for HD screens is the currently the "Twisted symphony method" (as he discovered it). I.e. You can use real Sega (or Namco) arcade IR guns (the ones with ten LED boards around the screen) with a PC by connecting it via a UHID from Ultimarc.
E.g. I'm. Using a Ghost Squad arcade gun to play Ghost Squad on the Dolphin emulator in this vid:
View: https://www.youtube.com/watch?v=8gx00X9qgzM
It could just as easily be a Time Crisis 4 gun. Those guns work on any display.
You can also use any positional arcade gun with emulators on any type of display:
View: https://www.youtube.com/watch?v=l-EdwvQJZlg&t=37s
There was an HD screen compatible gun for the PS2 called the EMS Topgun 3 but nothing beats a real light gun on a large CRT.