Apparently, my system isn't adverse to running AGP video after all. It just doesn't want to admit it's willing to.
There's a CMOS setting which lets you choose whether your primary video device is PCI or AGP based. The computer will function okay in AGP mode, provided you don't actually have any AGP based video cards installed in your system. Switch it to PCI mode, and AGP works fine. So I can have my dual monitors back, just not in anything close to an optimal configuration. But that's okay, for now.
I wasn't able to get much work done these past few weeks, because I had optimized my web editing tools to take advantage of two monitors, and with one missing, the software crashed. Hopefully, they'll work again now, and I'll catch up on a few things before pursuing a more permanent solution.