Strike "LH" from our diplomatic lexicon and simply say "The Hezbollah party of Lebanon," or better yet "Lebanon." Then make it very clear to all that when Hezbollah acts so does the state of Lebanon. Take them out of their quasi-state status and place them into a full state status. Now our state tools work. If the government or populace thinks that is unfair, well then they need to police their own problem; and if they can't then it is proven true by their inability to do so.
Everyone knows the state of Lebanon cannot control HA or "police their own problem." They do not have that capability, so how can our "state tools" work in that case?

Should we use that logic for the AQAM sanctuary in Pakistan, despite the fact that we know Pakistan is incapable of controlling its territory?