To be clear: you can not boot with one working videocard in slot A and the badly flashed one in slot B? Even after reversing them?
Status update: My BIOS does not have any such IRQ settings options, probably cause it is a gooshy GUI UEFI interface instead of a good old text based BIOS. So I decided to try it anyway.
I recall seeing a long time ago that the PCI-express specification allows for hot-plugging, but I still gave myself heart palpitations as I swapped the cards out. What happened was, I got everything working with 1 old PCI card and 1 working 5870 both connected, then I rebooted and hit F8 to cause the boot process to stop. Then, I unplugged the PCI-e connector on the good 5870. As I was readying a dead card to be plugged in, the machine decided to reboot, instead of sitting and waiting for me to insert the dead card. I inserted it anyway, and long story short it finally allowed the mobo to POST and boot properly while connected.
Once it was booted, it required 1 more reboot before the card was detected and a driver installed.
At this point, the driver is installed, and the card appears in the device manager, but that is it. I can't get cgminer to detect it, nor can I flash it with RBE. My next step is to create a boot USB with a commandline bios flash tool and a copy of a good bios and try that out.
TL;DR, 2112's method worked so far that the video card is detected and allows the mobo to POST, but I still have not been able to flash it.