OP, the barrel you're being offered, is no different than the one you already have.
IMHO, if the gunsmith can't match the barrel profile, he should be using larger diameter sleeves. They are available in just about any diameter required. The sleeve, should have the threads cut into it, should have a shoulder about 4 cm from the forcing cone about .030 in all the way around. Then, the original barrel should be only thick enough to provide a profile, not take up the pressure. This isn't a difficult set up on a decent lathe.
Your pics are nice and sharp, but you are looking at another disaster, if you insist on loading 125 gr, full powered 357mag fodder into the pistol.
The way this smith is relining, is a recipe for disaster. That is IMHO of course.
Rebarreling a revolver or semi auto, isn't difficult. It is meticulous though and needs to be done right. Smith and Wesson revolvers, have the same thread pitch in each caliber. I will look it up for my own information but I think they all use the same thread pitch and only the threaded length is different.
The crazy cost of getting the replacement barrels manufactured is the big issue. A used M39 or M59 S&W will go for $350 to $450, even in excellent condition right now. The number of eligible buyers, is decreasing every day. A new barrel, made to restricted length, runs about the same cost. Then, there is the fitting cost, shipping cost, etc. This will bring the value of the pistol up to $900+ to get it re categorized as restricted. Very difficult to justify the cost and cheaper to just buy a new pistol that meets the regulations. Problem is, the pistols made after 1945, can't be passed on to anyone, including family, unless they have 12-6 status or the firearm has been fitted with a restricted length barrel.
Somehow, I think the excessive cost of the replacement barrels is because of heavy mark up prices. I know what blank barrels cost. I can get a 25 in X 3/4 hammer forged barrel, delivered to my doorstep for $125. This would make up five round barrels or sleeves. Add another $100 for a CNC set up and $50 for fitting, the barrels should be economically made up and install for around $250 per firearm. Stainless would be a few bucks cheaper, not much more.