This is why a lot of us prefer stainless steel shot!
We've used shot that rusts but kept it in the barrel with soap and water covering the shot and it doesn't seem to rust. Cleaning is usually nothing more than running it with soap, change the water , add more soap and run it again . Repeat until clean.
I don't know about soap being bad for the barrel. My favorite is a powder soap mix sold at rock shops and Rio Grande. Seems to work better than Dawn. Maybe it has a rust inhibitor?