Common methods of welding be it with electric arc or gas flame will induce a certain amount of stress into both the base and filler metal. Any type of common welding would dictate the need for correctly heat treating the entire barrel following most any of these processes; even the most tightly controlled process will have resultant issues in the HAZ (heat affected zone).
Flame-spray does not offer a fine degree of deposit control and this process would require pre-heating the entire barrel evenly during the duration of the process to help prevent warping but it will not guarantee that it won't warp.
Cold transfer twin-wire arc spray offers several benefits. Because very little heat is induced into the base material, there is almost no chance of warping or inducing stress. This method also affords a very high degree of deposit control through the use of stencils and the air jet.
The biggest problem with any build-up method is obtaining deposit material alloy that exactly matches the base metal alloy; any difference in the alloy will show with just about any finish you put on it. If you can obtain the same filler alloy and the weldment is properly post-treated, it isn't a problem maintaining the same color, grain and hardness consistency throughout.
It is possible to do surface spot build-up without undercutting using variations of the GTAW & GMAW processes but it's not something that is easily learned nor accomplished to the degree that would be required for this application. In common circumstances, surface disruption of a few thousandths around the edges of the weld is not an issue but in this application they'll stand out like a neon sign. There are two ways to completely eliminate any undercutting and that's to overfill then remove the edge portions that are not fully bonded to the base material or to pre-heat the base metal to the point where 100% bonding can be obtained.
Cost in both time and materials is going to be the determining factor. Just because it "can be done" does not mean that doing it is "cost effective". Standard run on a cold transfer twin-arc spray using "cheap" consumables is around $450/hour, figure in the additional set-up time for fine control and special alloys and you're looking at some big bucks. Even going with a manual process like GTAW or GMAW the set-up and materials costs will be excessive and I can pretty much guarantee you're not going to find GMAW filler wire that is color compatible with any of the common barrel alloys.
Bit-o-history:
The theory of using inert gas arc shielding dates back to the 1890's but it wasn't very practical until the late 1920's. GAW (Gas Arc Welding) and specifically CAW (Carbon Arc Welding) was the first common manual process but not really in wide-spread use until the early 1920's. In semi-automated processes, some industries were using materials other than carbon electrodes but the use was pretty much limited to controlled atmosphere containments such as in the vacuum tube manufacturing industry. GTAW (sometimes called heli-arc) was most commonly used in aircraft and other specialty industries by the late 1930's and by the mid 1940's it saw some considerable advancements and was getting used in a lot more industries. GMAW came about in the 1920's but never became very practical for non-atmospheric controlled containments until the late 1940's and by the early 1950's it was making its way into a lot of industries and was in wide-spread use by the early 1960's. By the late 1960's GMAW was finding its way into smaller job-shops but it didn't make it into wide-spread use in the field until the late 1970's and at that its use was limited more by the unwillingness of humans to change their ways. During the late 1980's and through the 1990's, considerable advancements were made in both the GTAW & GMAW processes thanks to the considerable advancements in electronic controls. Reliable power-factor, timed pulse and waveform electronic controls opened up a whole new realm applications as did advances in SSFC (Self-Shielding Flux-Core) wire and the ability to package a wider array of alloys in wire form. Auto-feed filler GTAW also saw great advances in the same time period. Another misconception is that automated welding is a rather "new" idea when in fact it is not. Fully-automated self-sensing robotic welding is new but semi-automatic robotic welding machines have been in use since the late 1940's. While the fancy electronic controls and sensors were still decades in the future, some enterprising mechanics employed the use of chain & gear drives, cam followers and clutches to make extremely efficient automated welding machines. The early automated processes began their lives primarily in the boiler and ship-building industries where repetitive pattern welding is quite common. Despite all the advances in electronic controls and robotics, the pipe welding industry still relies heavily on the same basic all-mechanical semi-automated welding machines that were being used more than 60 years ago.
Some of my toys.