Hard to tell from the photos if this will work with your barrel but you can give it a try.

The first attempt to remove rust I use is scraping it off. Some guys insist on using softer metals like an old penny. I generally use the flat back of an old pocketknife blade. Yes it's steel which is as hard as the barrel and the potential is there for scratching but I've never had much issue with that. The edge is not sharp, it's slightly rounded and basically crushes the rust off. It does not harm the finish where their is no rust and where there is rust, it usually leaves an appearance very similar to the unrusted areas. I have a lot of posts on here with before and after photos so you can see how they turned out. This only really works if the metal is not pitted or at least not pitted much or deep. It's time consuming and labor intensive but it is also very satisfying watching the steel come back to life.

If pitted, I go to method two which is electrolysis. They say this does not remove the existing finish but in my experience it does to a degree at least. Another method I used because I did not have access to an electrolysis set up (they aren't super expensive, about $40-$50) is soaking it in white vinegar. This removes everything to bright metal with the exception of deep pitting in which it turns the orange rust to a more stable black rust. I don't use this method anymore with the exception of small parts. But in your case with the rifles value already gone so originality is no longer important, you could to this with the intention of re-bluing afterwards.