Well, you are certainly considering the old school approach. The new barrel gives you the highest probability of getting a good result that lasts, but I'll describe a bit of what's involved in the old method so you know what you'd be getting into.
Getting a barrel weld-up is how it was still being done when I started learning to build match guns in the early 80's. It used to be the standard method on carbon steel barrels. I've seen a picture of the late Bob Day doing the welding with an oxy-acetylene torch, adding wire from a magazine spring as the filler. I sure wouldn't want to try that myself for fear of getting the flame oxidizing and carburizing balance wrong and the barrel too hot. It's a skll. The point is, the filler matters. It has to be hard enough for the service but not so hard you can’t scrape the fit. The fly in the ointment with your Randall is that it is stainless, and the galling issue will come up. Since stainless barrels with oversize hoods and lugs for fitting are made and work fine, I’d suggest looking at barrel steel rod for the filler. Your welder may have other suggestions?
Hallock's .45 Auto Handbook (1980) has a photo of a welded-up barrel on page 166. but it isn't very clear. He does, however, have clear line drawings showing how the shaping and scraping are supposed to come out on page 169. You should see that if you are going to try this. Brownells sells a cutter system that works in-situ, but it is too much expense for doing just one gun. Filing and scraping will be reasonable for that. The late George Nonte, Jr.'s book, Pistolsmithing, has a clearer photo of a weld-up barrel ready for fitting on page 359, and a line drawing on page 367 showing the weld filler positions. He shows the weld buildup including the sides of the hood (barrel extension) their full length. Welding the width of the hood is necessary, but you only need the sides done far enough to guide the hood in its channel in the slide, so half way to the corners is fine if you don’t mind the appearance.
Nonte labels the picture an arc welding job. He meant heliarc (TIG), I am sure. A friend of mine with a stick welder once insisted he could do the job with that equipment. Rather than hand him a barrel, I had him try to prove it first by building some weld up on the edge of scrap piece of 3/16" steel plate. It didn't go well.
An important part of the welding procedure was getting the chills right. Guys who did it frequently had machined copper chills that hugged the chamber and made good electrical contact for the welder ground and spared the chamber from heat exposure or ground return hot spots. You may be able to improvise by splitting a slightly undersize copper pipe and using barrel vice jaws to force it to conform. Boring a chamber O.D. hole in a sandwiched pair of aluminum bars would be easier. The old time gas welding trick of submerging all but parts to be built up in wet sand is an alternative.
Usually, before fitting a barrel, you would have fit the slide to the frame of a carbon steel gun, mainly by peening the frame rails down. Many gunsmiths recommend you don't even try to bring the frame rails down on a stainless gun. The galling issue and the fact stainless is less ductile than carbon steel make it very risky to the health of the receiver. Assuming u don't try, the barrel will wind up rather high when it is locked into the slide lugs.
One short cut you could try is the method Bob Chow used to remove vertical barrel play. He cut a piece of spring steel just a few thousandths wider than the rearmost locking lug recess in the slide, tapered its edges, then wedged it up in the recess. Apparently these oversize bent shims with feathered edges didn't fall out. They centered the chamber axis vertically over the firing pin tunnel, and the rounded shape tended to center the barrel left to right as it moved up into battery. It did the same thing the Cash Gustin screws do, but without altering the frame. The necessary thickness of this shim would depend on your existing lockup free play?
Have fun, and be careful.
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