Last time I spoke to Briley about this service they were sending it to Aaron Little.

Spoke with Aaron and he is $4000 and a two year wait and the barrels will have no choke unless you install screw in chokes.

If a competent gunsmith measured the thickness with the pits and determined them safe, I’d be inclined to just live with them and hunt with the gun. There must be a story there. At first glance, it looks like corrosion caused by blood to me.

Damascus or Steel?
If damascus Peter Dyson has raw barrel tubing which I believe could be made into barrels, however to turn, bore and make new barrel wouldn't be inexpensive.
Good luck

Barrels are steel.

Did Jim think it was safe to shoot as is? I'd be inclined to leave it as is if so. I know that the guns you post are always in pristine condition.
The Eyester's are well known for their barrel & choke work in this neck of the woods.

Jim Eyster and I went over the gun thoroughly and we decided to have the exterior of the barrels Micro welded. That has been done and Jim was very happy with the work. We are going to go over this gun from top to bottom and try to put it into the best possible condition. The welding shop specializes in gun work and said they have great success with invisible repair after rust bluing. This is such a nice gun and so rare it deserves some special treatment. I totally trust Jim when it comes to barrel work and he says they will be totally safe when he is finished.

Make sure to show pictures of the finished gun

Some pics of microwelding done at a friend's high tech shop. He's not in the gun repair biz and did it as a favor. The pic collage shows my start of cleaning the pitted areas ….. through the finished rust bluing. The left barrel muzzle was the worse and second pic is one I sent with the barrels. I left the choice of filler to his welding tech and told him about the soft solder holding the ribs. He said not to worry; he'd take precautions. The microwelding was also done in other places on the barrels not shown in pics 2-5 and it didn't affect the solder joints anywhere. I did the striking and finish polishing myself and was careful to keep a sharp edge at the muzzles. Just to show what can be done.

Frank, thanks for positing the pics. Very nice results!