Y'all keep me laughing with the "brass thingy from the rifle range"! And yeah, a .404 Jeffries would be ideal.
I have run into the same problem of soft metals clogging the file teeth. I get used stainless dental picks from my dentist, and run the tip along the edge of the teeth - across the file. Gets rid of brass, aluminum, etc., fairly quickly. I will, however, have to try that trick of freezing the file then carding it!
The picks can also be quite useful in getting rid of dirt from 90 deg corners and other tight places. Use lightly, so you don't scratch what you are working on.
For me, my 9.3 x 62 mm brass is about as British as I can get. Well, South African anyway!