Yes, you can swap the 2GB Radeon HD 6970M for the 1Gb version and will probably not even notice the difference (depending on what you're using your iMac for).
Since you're comfortable enough fixing your own iMac, and you'd have to get to the GPU anyway to replace it, I wouldn't replace the GPU at all. Instead I'd try fixing it by reflowing the solder. To do this, you need to:
- Disassemble the iMac enough to remove the GPU
- Set your home oven to 205℃ (400℉)
- Place a sheet of baking paper on an oven tray and place the GPU (only the GPU itself) on the oven tray
- Place it in the oven for 8 minutes
- Remove the tray and let it cool
- Remove any old heatsink compound from the chips and replace with new compound
- Reassemble everything
Depending on how familiar you are with disassembly/assembly, I'd allow anywhere between 3hrs to most of the day for doing this. Also, you will require some tools, but if you don't have these you can easily purchase them online.