If 食って掛かる is one word meaning "to lash out at someone" or perhaps closer in figurative language, "to bite someone's head off," how should I think of the 掛かる? If I want to understand the logistics behind this word, perhaps I can imagine it as "to be hit with someone's bite," wherein 掛かる provides the "to be hit with" meaning? Is this the right way to think of 掛かる here?
掛る means "to be hung on, get involved into, bite into, get locked, lean on." 食って掛かる's literal meaning is something like "bite on and get locked on it (for fighting mode)".
Obviously, 食って掛かる is not a single word. It should be called an idiom.