If you have a vertical mill but no rise and fall cutter, you could make a long stepped series of cuts with an end mill around the "circumference " save a lot of the filing, and get consistent dimensions.
Clamp the barrel on the table against an end stop. Set depth, run the table in and out to make a cut transverse across one flat of the barrel. Rotate the barrel and take another cut etc. Traverse the table, reset depth, repeat.
You'll then have a series of gentle steps to blend by draw filing. But a lot less filing and you won't be eyeballing so much.
The barrel will be resting on the intact flats at each end. As you progress, the barrel will become unsupported across much of its length. You'll want to feed slow to minimize deflection and/or stuff shims under to support it.
No, I haven't done this before. I am literally sitting in an armchair playing gunsmith.