Something I've been thinkin' bout here...
Okay, so I've been contemplating this one more, and I wish that you, me, and it was sitting in the shop, so we could team up on it.
This welder uses two separate cores, and they're wired with two primaries together, hooked to the "B" phase, with the far ends of each hooked to A and C respectively. This corresponds to delta, but with one core (A to C) missing.
The outputs are wired just the same... tied in delta to a three-phase rectifier array.
My gut feeling here It will probably work alright wiring A and C of the primaries together, but it'll probably work the rectifiers over pretty good, as they'll be carrying current in a different sequence, so the cyclic currents will likely be high.
If you REALLY wanted to try an synthesize the original working condition, you could try driving one coil directly off the mains, and driving the second coil with some capacitors to phase shift it. Optimally, you'd do it at a 120-degree interval (360 / 3 = 120), but the limit would be 90 degrees.... I'm thinkin' that the end result would be fine at 90, you'd have quadrature going through the rectifiers... which is essentially what happens in an H-K setup.
Off the top of my head, I'm thinking that just installing a pair of caps in series with each lead of one coil would be enough to determine wether improvement is within reach. The mathematics for estimating appropriate capacity will follow the lines of what I did with the CP and SRH's, so get the ballpark, and then give it a try.