Hi guys, thank you so much for all the advice! I spotted reels of waxed cotton which would avoid the fuzzing, but probably won't take paint. I tried the twisted wire trick - I did that with my micro Dremel but its speed control isn't activated until it's already on so it ran out of control before I could stop it! The cable does look spot on though, just got to remember to press stop...
It's actually to go on the cable reel underneath the Tamiya FAMO - they do supply a reel of white but the stuff in the Amusing Hobby kit is somehow more realistic and stiffer. If I can get a long enough run of copper wire to braid then I might well go with that instead.
But cheers all!