I'd just soak the rugs in white vinegar for a few hours. Put them in a tub, pour it over, and let them sit. Then wash. Works for puppy issues on carpets anyway.
You'd be surprised at what all white vinegar is good for. If you ever need to clean stuck-on-gunk inside a bottle, put some baking soda in the bottle, then pour in vinegar. Will boil it off.