It sure looks like I was somehow using the 7.82 gigatonne number as the total, rather than multiplying it by the necessary 422.

A simple calculation in units(1) shows that your figure is the right order of magnitude:

  You have: 400ppm 4pi earthradius**2 atm / gravity
  You want: trillion tonnes
        * 2.1080571
        / 0.47437046

That's a bit low because the 422 ppm number is by volume, not weight, but that is relatively easy to correct with the molecular masses, assuming ideal gas behavior:

  You have: 422ppm (carbon + 2 oxygen) 4pi earthradius**2 atm / gravity (21% 2 oxygen + 79% 2 nitrogen)
  You want: trillion tonnes                                     
        * 3.3925982
        / 0.29475934

And that's within 3% of the number you give from Wikipedia.