Very wise words dZ!!
In my job before retiring (metallurgical engineer) I knew the guru of this cryogenic process and our company did extensive testing on tooling (supposed to last longer) and gear wear (supposed to last longer). The gears treated definitely
lasted longer! The tooling did not show any improvement. I don't know why it would make a barrel more accurate. It "should" make them last longer but a 4140 steel barrel might last longer while a stainless (less ferrite) wouldn't. If you're going to try it, test it. Don't take anyone's word for it working or not working. Including the salesman.
my thinking is that anything trapped underneath the copper can cause corrosion
Just something for thought. If something is trapped underneath copper it is no longer exposed to the air (H2O). Therefore it shouldn't cause rust since it takes oxygen to cause rust or ferrous oxide (FeO).