Multi-monitor problem, possible fix?

Working with an older AMD gaming computer (temporary HTPC, lol) and two displays ...

Asus M3A78-EM mobo (780G+SB700 chipset) with 1 HDMI output (not used)
Radeon HD7870XT GPU with 1 DVI + 1 HDMI outputs
Windows 7x64 (natively supports DirectX 11)
AMD Radeon "Crimson/Omega" Driver 15.3
MPC-HC media player software
Chrome Browser

LG W2343T monitor (23" 16:9 1920x1080 60/75Hz, connected to GPU through DVI)
Samsung TV (50" 720p, connected to GPU through HDMI)

I'd like to display desktop/computing stuff on the monitor but have the option of displaying movies and netflix on the TV (while still using the 23" monitor for desktop/computing stuff). Can this be done? If so, how? Am I doing something wrong or do I need to install better software? (Never been forced to work closely with such a mismatched hardware selection before, lol.)

The WinOS seems to only allow both displays to be used as a multi-display desktop or as a pair of identical desktops, but doesn't allow an application (like a movie player) to launch on only one screen.
While the AMD EyeFinity stuff seems to require both/all displays have identical resolution. (I can choose to set my monitor to 1280x720 or have 1/3 of my image cropped off on the TV.)

You'd need to set up a virtual machine to get this to work. You could install another OS and run it as if it's another machine. As far as I've ever been able to find, this is the only way to do it.

Good call. It didn't occur to me because the hardware on this machine is so ghetto, I'm not sure how well a DDR2-based Athlon II X3 could multitask, lol, but I'm sure it would take a noticeable performance hit.

Yeah, but you might get away with it using older software. Keep your normal OS on the one machine and make the virtual machine run an older version of XBMC on Linux for your home theater needs.

Running linux Mint (Rebecca/Xfce) and a couple of Win7x64 instances under Wine, one WinOS running Netflix/etc on TV screen (through IE 8 and Silverlight 5), the other WinOS running desktop/etc on monitor. Surprisingly still getting almost 60fps on DX11 games and benchmarks, lol, while simultaneously streaming HD 720p video . But it seems like a whole lot of overcomplicated kludge, all designed to workaround limitations built into the Windows OS and drivers.

You can try installing another OS and check if your multi-monitor working.