aamoroso,
"I don't have a ton of faith in that calculation" I liked that.
I do not think that the calculation per se is dubious, but the inputs probably are. Those numbers are sales numbers and are usually on the high side.
The figures usually assume that your mains voltage stays constant even if heavily loaded (a welding tranformer takes all it can from the mains - and then some). It is usually also measured as a "short circuit current" which means that cables, tongs and the actual plates that you weld are excluded from the calculation. A simple test will show if your mains is tough enough; connect a lamp and see if it "browns out" during the weld. If it does, you really have to do something about the mains.
It is not uncommon to find half of the total voltage drop in the mains and the transformer and the other half in cables, tongs and welding material. If there is a rectifier, you usually have one third in mains and transformer, one third in the rectifier and one third in cables etc. So anything between your calculated current and half of it is possible. I would guess that you get between 60 and 80 percent of what you calculate. But I cannot guarantee it. Measure it.
There are very few 20 kA current transducers around, so it could be a problem to measure that current. Look for rogowski coils. They usually do a good job at these current levels, but be prepared that some of them cannot measure DC (If your current source is DC, which I do not know).