What OS are you running -- El Capitan (10.11.6), with the patched boot.efi?
According to
the MacVidCards FAQ, the 7970 should work on a cMP 1,1 --
if you are running Mavericks or newer with a patched boot.efi (thanks to the work of tiamo, pike r. alpha, and many others over the years).
EDIT:
Now I realize you are saying the card is not recognized in Slot 1, only in Slot 2.
By default Slot 2 is configured as x1, so that makes sense.
Take a look at
this thread for someone who had a similar difficulty to what you describe.
The cMP 1,1 can reconfigure its PCIe lane widths using the Expansion Slot Utility, found in /System/Library/CoreServices.
More info here.
Good luck! I have a 1,1 that runs El Capitan (10.11.6) flawlessly with a 5770 GPU. I could test it with an R9 280X if necessary to confirm that it works with a card very similar to your 7970.