Hey aerowelder,
I'll try to give you some information to help you have some success with your project. As a retired Master Gunsmith, I know the making of those old musket barrels were done several ways. Most common method was the hammer-forging process of the metal around a bore mandrel. This method resulted in some irregularities in the overall uniformity of wall thickness and, at times, would cause a failure(rupture)....quite common in those days. The barrels were also quite soft and nothing near the strength of turn-of-the-century improvements innovated for the high-pressure center-fire cartridges needed for the military.
Anyway, just some history to allow you to understand the metal you are attempting to repair. In May, I had the opportunity to do that exact repair on several muskets that were to "somewhat" be restored for display purposes only. The barrels were quite deteriorated with rust & corrosion and the owner said all he needed was to have the holes sufficiently filled & dressed off and an application of "rust bluing" to conform to the original finish & color. The customer had the muskets disassembled & I removed the barrel nipple to insure the powder chamber was clear. Since both were .69 caliber, I only had to make one aluminum barrel bore "slug" to reinforce the weld area & keep the weld heat control to a minimum. The "slug" was drilled & tapped for a 1/4 X 20 thread, turned on the lathe to fit the rusted bore & a bit(.020) undersize, cut to a length of 2", & screwed onto a steel rod. This would be used as a heat-sink & backing to allow the weld to have a surface to "rest" on. I lightly dressed the weld area with a flap disc just to clean the metal bare so the weld had clean metal to bond. I used my Miller Sidekick 90A MIG with ER70S-6, .030, & C25 with the heat setting dial on 1 & wirefeed dial at 4. I had to very carefully just "tack" the edges of the barrel hole in varying location to get a solid border & even as careful as I was, I had several tacks burn-thru because of the deteriorated metal. I finally got a solid border & carefully continued the alternate tack method to completely fill the hole. I let the barrel cool in sand to insure no cracking and both repairs came out quite nicely. After gently dressing the welds with my 1" belt sander almost to the finish diameter, I hand-filed the balance of the weld to the barrel diameter & hand-sanded to smooth the surface. I cleaned with acetone & warmed the repair with a propane torch & applied Birchwood-Casey Rust-Blue to match the color with several applications and they came out quite nicely. The customer was very satisfied and did not hesitate to pay the price I indicated prior to starting($125/ea.) when I explained the parameters to get a quality repair he was seeking.
Oh, BTW, these 2 muskets were going to be displayed in a local restaurant dining room that was totally decorated with photographs & memorabilia of the Old-West era & quite detailed with historical facts. AND! I got a compliment great dinner for myself & lady friend for my effort. Anyway, just take your time, do your prep homework & procedures steps to insure success. Do some practice welds on very thin material to get your heat/wf settings to an optimum and don't think you will be able to run stringers....not gonna happen....tacking only. Good Luck........... Denny