To answer your question: No, it is not possible to set the Windows System Locale for a single user.
About running as a non-admin user account. That answer is probably answered with XP in mind, where it used to be a Run As option (instead of the modern Run As Administrator) where you would be able to enter the user of your choice. You can still achive it nowadays by holding the Shift key while right-clicking the program, and there will be a Run as different user option.
Now for your problem: You may be able to use Microsoft's AppLocale program to accomplish that.
It was developed for Windows XP and isn't supported for later versions of Windows, but as outlined in this guide it still works at-least on Windows 7 Using AppLocale for non-unicode programs in Windows 7. You can download the program itself from the Internet Archive version of the Microsoft download page.
Another solution (which is officially supported for newer OSs up until Windows 10 is the Locale Emulator http://pooi.moe/Locale-Emulator/