Rhodium plating, when used on this type application, is a thin, hard, shiny, durable, whitish, metallic protective coating mainly used to prevent tarnish of the base metal. Due to stress problems with thicker deposits, it is usually only about .000005" - .000030" thick, more or less. That's approximately $0.06 - $0.36 worth of rhodium per square inch of plated area. It's one of the most inert metals on the planet and I know of no electrolytic solution that will dissolve it. Aqua regia won't even touch it under normal circumstances. To remove it from the less expensive jewelry (silver, usually) or white gold (which contains nickel, which some people are allergic to), on which it is often applied, it is usually buffed off. The jeweler doing this doesn't worry about recovering it, since it has little value at the thickness it is plated.
I would say that the scrap value of those badges would be primarily in the base metal. The purpose of the rhodium is not to add value to the piece, but to protect it. Even if you had a ton of those badges, you would have a very hard time finding a refiner that would pay you for the rhodium. It's just too difficult and expensive to refine and the majority of refiners wouldn't be set up to refine it. Rhodium refining isn't something a hobbyist refiner could just jump into. It requires a lot of knowledge.
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