I welded a damascus barrel that had a hole very similar. Maybe not as large in diameter. I ran an argon purge through the barrel. I then placed the tip of the 3/32" diameter 70S3 filler wire over the hole. I used a 3/32" diameter tungston that I ground a slightly blunt point on, held it over the wire and hit the pedal for for what was probably no more than a second. It fuse perfectly and was nearly undetectable. It took very little polishing on either side. I was most concerned with the inside finish. You can't put a mandrel in because you can't get full penetration and not have purosity cause by the disimilar material of the mandrel (usually brass or aluminum) as you weld into it. The purge is the only way to have a perfect weld inside. The trick is to get fusion and not drop a large dimple into the ID of the barrel that would be very hard to polish. The deeper into the barrel the more difficult the polishing. I used a 3/8" diameter rod with some emory cloth to smooth out the dimple.
Bill G.