Best GPU for a weirdly unpowerful setup?

So we are setting up a broadcast studio at my school. We have a Blackmagic ATEM TVS and we're gonna use MXlight to stream the broadcast live. We also have to run ScreenMonkey to play videos into the mixer via HDMI in real time. In total, the PC will have 3 displays, one being a touch display for the BlackMagic control software, one being a non-touch display for ScreenMonkey, and the 3rd being the actual screenmonkey output into the mixer.

The machine is a Core 2 Quad with a WD blue drive, and 4GB of RAM. What would be the best GPU to do this, and should we try to upgrade the RAM?

I'd say that the RAM should definitely be upgraded, somewhere around 16 GB, as I'm taking it a school isn't going to want to spend too much money, but at least you'll get better render times. As for the GPU, Quadro's are nice, but I use a GTX 760 personally, and although I haven't gotten the opportunity to stress test it yet, I have been happy with the results. For the Hard Drive, what is the capacity specifically? I ask because I assume the school might want the footage archived.

I'm not too worried about the HDD, we will only store a small amount of footage on the machine itself, the "archives" will be on our main editing rig which has plenty of space (and dual Xeons, yay), so capacity isn't a big deal, but will speed be a big deal? The toughest things its doing is ScreenMonkey and Mxlight, but I think mxlight works mostly the GPU and the network, and I've run ScreenMonkey on a Pentium with a $30 GPU and 4GB of RAM before.

So would a 750Ti work? I like that we wouldn't have to worry with a better PSU for one of those, I'm not sure how much wattage ours has now but I'm sure its not anything fancy.

The R9 280 is around the price of the 750 ti and offers a lot more performance. 

Find out if the software you use is either opencl accelerated or cuda accelerated, if yes to one of these then pick that one.

gpu acceleration will make the machine feel a heck of a lot more like a modern machine :).

as for ram, if any of the machines are doing local rendering you want to avoid scratch disk as much as possible so max that out within your budget.