Matt, if you make your boring bits and reamer out of modern tool steel, either water or oil hardening, I'm sure they would cut the modern steels used to make the barrels.
Building the boring bench and making the traditional tools would take a bit of time and effort, but I don't think that would compare to the time and effort it will take you to make the two shotgun barrels for the double.
Most people that make forge welded muzzleloader barrels today use wrought iron skelps. Obviously, modern steels like 1018 and hardening steels like 1082 can be forged welded because knife makers do it all the time. To me, the challenge would be making a full length shotgun barrel.
Here is an interesting video of a guy forging a damascus twist pistol barrel from the modern steels mentioned above. His barrel is only a few inches long, but notice all the work he puts into it, and he's using modern power hammers.
He has several videos on Youtube of different pistol barrels he's made.
I'm not sure just how traditional you're trying to be in this build, but how are you planning on shaping the exterior dimensions of the barrel? The 18th and early-19th century method was with very large grinding wheels. I think this image is from 1850s Birmingham, England. This is how the English got such thin barrels towards the muzzle.
The image below is from 17th century France and shows a different method where the grinding wheel was smaller and located below the barrel on a bench similar to the boring bench. The barrel could be moved lengthwise over the grinding wheel and possibly rotated like on a lathe. They could grind octagon flats as well as the round section of an octagon-to-round barrel with the same setup.