It seems like the 64bit edition isn't that supported for some reason. Integral edition and extended updates /modifications seem to target 32bit version. So which version would you suggest or any way to make xp recognize the RAM capacity fully without crashing
If you have 64bit applications that take advantage of the extra RAM, go with the XP 64bit edition. If not, XP 32bits is enough. You can use PAE patches to allow the use of the extra RAM, if your hardware and drivers are compatible with it. The modded editions like the "integral edition" you mention includes additional software to help you take advantage of newer hardware with XP.
If your PC has this processor(CPU) and has 12 GB of DDR3 RAM, it can run Windows 7 64-bit or Windows 10 64-bit. If it has a solid state drive(SSD) and a dedicated graphics card(GPU), that is even better.
I know I just use it as a "Retro" PC for XP and 7 Programs/Games on a small 128GB ssd I have 3 systems in total main one has LTSC 2021 and 5800X3D