I went with a low power graphics card, the Radeon HD 5670. My PSU can handle it, so no need to swap that out.
As soon as the RAM arrives I'll sit down and take another gander inside, perhaps with a scissors to cut the plastic bind that holds all the wires together in that unholy bulge. Set some time aside and clean the place up a bit.
I've been amazed with the response to this thread. I didn't think I'd get so much help so quick!