The problem is hoop stress..
Pressures should be around 18,000 psi.
Heating only a portion of the hoop is going to induce internal stresses to the steel at only the point of welding (HAZ)
The barrel OD is the same, but after reboring and rifling, the welded barrel will be thinner (as was an original .45 barrel). YOU DO HAVE TO WORRY ABOUT THE PRESSURE THING...
Hoop stress is always there, reducing thickness of hoop (and welding on a portion of it) DOES NOT MAKE IT STRONGER, IT ONLY WEAKENS IT.
Add in OLD unknown steel type, and batch number, and certifications.
No certified welder should weld that, without an engineer approved welding process. Welder would be liable if it failed, and killed/maimed 1 or 2 bystanders.
Shotgun pressures are a LOT lower, (as is the quality of the steel) welding on them should be treated in the same DO NOT DO IT, way. I have seen it done, and still do not recommend it.. You are playing with LIFE AND LIMB.
No National Proof House would allow it.
It is an antique. Retire it, or sell to a collector. Remarking an original barrel could be considered counterfitting.
Normally a liner MIGHT be considered, NOT enough metal on that gun. Add in it being a large bore revolver with a high stress, threaded, necked down area, where barrel screws to frame. This negates any chance of lining..