I was able to test the 5D Mark III, but even after hammering on the camera for a few days I was unable to get the error 80. Using your steps to reproduce the issue are you able to get any other card (other than Eye-Fi) to produce this error? From reading some Canon forums it seems that the error 80 happens to some and not others. I was running the latest firmware (1.1.2) and I was using all Eye-Fi card models as well as different firmwares and still no error. If it happens to be appearing on a camera by camera basis then I would assume you would be seeing this error on a non Eye-Fi card. Would you be so kind as to try and let us know?
I should be getting my hands on another 5D soon, and I will be noting the serial numbers of each camera I try, but until then it would be good to know whether the errors you are seeing are specific to your Eye-Fi card.