All versions of Windows XP Home are 32-bit.
It is highly unlikely that the presence of Service Pack 3 is playing any role here (but I suppose anything is possible).
What is the name of the software? Have you contacted that company for support?
It sounds like you have three PCs: an old 32-bit PC with Windows XP at the out-of-date SP2 level, a newer 32-bit PC with XP Home at the proper SP3 level, and a newer PC with Windows 7 (which is 64-bit). Which version of Windows 7 are you running: Home Premium, Professional, or Ultimate?