Not a dumb question, just there is not a good answer IMO. I don't use those mops for the reason you identified.
I either squirt sauce on directly or use fine silicone bristled pastry brushes
EDIT: For thin mops I have been known to use a sock (yes, really) or cloth napkin. Crumpled and held by tongs. I don't use those much, because I do not thin mop much, and when I do, it would usually come from a squirt bottle.