Thanks for the 'heads up' about getting the barrel too polished before browning- my last effort was left 'in the white' ("Armory Bright?") per the instructions of the end user (a.k.a. my Chief of Staff and She- who- will- be- obeyed). So I polished it all up and it looks very pretty.
To do this, i draw filed, then used wet / dry sand paper (any automotive supply store- NAPA, CAP, AutoZone etc.) in decreasing grits down to about 800 grit. You can get a mirror finish if you like.
USE A BACKER on the sandpaper, or you will get dished areas. You may also want to put a piece of stiff leather between the wood backer and the paper.
Good luck!