I do not think this has been TIG welded. Why? Because any TIG welder with any competence at all would have left these pits filled. They would no longer be pits. If they had been drilled holes, a competent TIG welder would have completely filled those holes, although there might be evidence on the inside wall of the barrel.
I would still want to get a good look at the inside with a borescope before proceeding. Also, file off that bloom of corrosion around the pits with a needle file, clean them out with a dental pic, and get a measurement of the depth. That looks somewhat similar to the pitting I spoke about in one of the rust bluing threads on an otherwise very nice set of Baker Damascus barrels. It was very localized and deep, actually going all the way through the left barrel, and ruining the top rib. But the surrounding area was unscathed.
