Is the shotgun a single barrel or a double? what make and model.
MIG is an acronym for a process; metalic inert gas. means that the electrode is consumable (the wire) and is fed automaticlaly while maintaining an arc and is sheilded by a type of inert gas, C-25, M2, etc. these are different concentrations of argon, CO2 etc. MIG is used for sheet metal but many of the MIG processes have high amperage and hi penetration for heavy duty applications up to and including pressure hulls on submarines. There are a number of MIG processes; short arc, spray arc, pulse arc and more. I was a structural and nuclear welder on Trident submarines for three years and we used all of these processes.
Th eprocess that you are thinking about that is suitable for gun parts is TIG; tungsten, inert gas. The lectrode is not consumed, it establishes an arc and wire is fed into the puddle and it is all shielded by a mostly Argon mixture (95% argon) with this process you have excellent penetration, can control the heat and puddle and keep the weld small with lots of control. There are guys here who can recite all of the technical aspects of TIg, I am going off of my head and thirty years ago.
But TIG is the way to go for nay serious welding on firearms due to the small heat affected zone, control and penetration.