Now I'm no expert on magnetics, but the idea seems feasible - especially given Diver's post above.
As far as I can see, you'd have to design it as follows:
1. Size the primary turns and core to stand the maximum input voltage (from the charged cap bank) for the expected maximum duration of your weld.
2. Select secondary turns for a sensible maximum output voltage, and wind using the biggest wire you can find.
The catch could arise in that if the weld current did not flow, due to a high-resistance joint or something, then the transformer would try to support the full maximum input voltage and quickly saturate - probably blowing the primary. So you'd have to have a way of disconnecting the primary after a set period so that the core could reset.
It's interesting that the kWeld makes a point of applying a known energy to the weld, which of course results in a known temperature rise in any given weld area. But, of course, a capacitor charged to a known voltage also contains a known amount of energy so, to a first aproximation, you'd think the same could be acheieved by simply altering the charge voltage (as I presume John's unit does).
The difference would be due to the time over which the energy was spread and the rate at which the surrounding metal could conduct it away. I wonder how significant this would be?
Also, I would have thoguth you'd want to keep the output cables loomed closely to minimise inductance and potential harm from the generated magnetic field (you'll have seen mikeselectricstuff and the destruct-o-tron?).