Skip the dremel. For the barrel, since you are going to take it "to the white"- go get some wet-r-dry sandpaper. 200 grit and 400. Clamp barrel where you can work on it hands free (wooden pads in jaws of vise)
Wet barrel with light oil. Use the 200 grit in a shoe shine rag motion. Keep it moving, do not sit in one spot. After you have buffed ALL of the barrel to as good as it gets, wipe it down, reoil, and start with the 400 grit. This time, back and forth, lengthwise. You are removing the 200 grit scratches.
When you have it as good as it gets, would stop there. You CAN go to higher grits, but I would not for a shotgun. Good wipedown with clean soft cloth- you are now ready to degrease, and blue.
There IS a cloth (muslin) wheel for a bench grinder- you apply jeweler's rouge to the cloth, and buff with that- BUT- there is a learning curve here. The barrel WILL be snapped out of your hands and sent clattering across the room a few times. Unless you are doing a lot of guns, stick to the hand method.
What we have here is... failure- to communicate.