I carry a bandana for the kitchen (dishrag, pot gripper, water pre-filter), and a bandana for everything else (hat, body washrag, towel). I wouldn't want the kitchen bandana treated. I don't know if permethrin can leach. I might end up treating the everything-else bandana, but it gets cycled between wet & dry often enough I doubt any treatment would last long.
The local Walmart has the cheap ($1.50) one-sided bandanas, but they're big enough to tie around your head as a dew rag (and great for keeping ticks out of your hair).