Welding the top of the barrel WILL warp it upwards, no attempt at keeping it cool, or a little at a time, is going to stop it. This is a physical law of welding. As the molten area cools, it shrinks causing distortion, and also stress.
In addition, depending how fast it cools, the heat affected zone will change the hardness and ductility properties of the steel. Rapid cooling results in hard and brittle metal, while a slow decrease will make it soft and ductile.
Would you want either of these to happen to your barrel?
Oxidation (scale) on the barrel interior is also possible, as mentioned already.
35 years of welding experience is recommending you do not repair this by welding.
After looking at the pics, I would forget all about this gun.
There are other fish in the ocean.
In addition, depending how fast it cools, the heat affected zone will change the hardness and ductility properties of the steel. Rapid cooling results in hard and brittle metal, while a slow decrease will make it soft and ductile.
Would you want either of these to happen to your barrel?
Oxidation (scale) on the barrel interior is also possible, as mentioned already.
35 years of welding experience is recommending you do not repair this by welding.
After looking at the pics, I would forget all about this gun.
There are other fish in the ocean.