Wood block with slightly rounded corner and successive grits of silicone carbide paper. Just push up onto transition as you polish the flats. Don't go crosswise, or will be real hard to remove residual scratches

When I file barrel flats I do so with the barrel on a flat surface and wood blocks on either side that are almost the same height as the barrel flats. This keeps the file from tipping and keeps the flats flat.
When final sanding the barrel flats I use an aluminum plate that's 8.5"x11" and spray rubber cement on the plate to stick my sandpaper sheets to it. Then I hold the barrel on the paper and move the barrel instead of using a block. The large surface allows me to sand a long time before replacing worn out paper, and it's very easy to keep the barrel flat on the plate and not round over the flats as can happen when sanding with a block and paper.