I'm not a welder by trade, but have managed to do some fairly intricate TIG welding on thin materials. However, I would not even dream to attempt what you are proposing, and don't know of anyone who could do this with any guarantee of success. Laser welding is another process used for intricate repairs on thin materials. But while it is often used to fill and repair small pits or cracks, I have not seen it used to build up a relatively large area.

First off, at only .006" of wall thickness, the area you wish to build up with TIG weld is only slightly thicker than a piece of standard printer paper, which averages .004"- .0045". So you would have to be capable of welding paper thin steel without burning through. At minimum, a close fitting copper plug would have to be made to avoid any build-up of dingle berries in the bore. You would also have to fill the bore with inert gas or take some other precautions to prevent scale formation. You would presumably also have to do this over a relatively large area in order to subsequently strike the repaired area, in order to blend it in to the surrounding areas that have what is considered normal and adequate wall thickness.

Then there is the question of HAZ (Heat Affected Zone), as mentioned by RWTF. It would be difficult to know the effect upon the metallurgy of the repaired area without costly testing. And temperatures high enough to anneal or stress relieve would certainly be higher than rib soldering could withstand.

I am curious how you or Gunter Prommfer could be certain that this gun passed proof after the barrel was struck this thin? If it did, that's pretty amazing, and illustrates why the frequent posting of Burrow's bursting formula on this forum has no bearing on barrel wall strength. A closed steel tube .006" thick at normal pressures found 12" from the breech would pop like a child's balloon. I've seen enough hydro-testing failures of much thicker tubes to be confident of that statement. It is interesting that it hasn't bulged or blown just using standard loads.

If I was in your shoes, I'd be setting this gun aside and start hunting for a set of orphan barrels that could be fitted. But having once experienced about a half hour of total blindness from getting hit in the face with powder gasses, maybe I'm just being unreasonably cautious about these scenarios. However, I can promise you... a lot of shit goes through your mind when you suddenly lose your eyesight.