Sounds like the neighbor has a "militia-fear" thing going on. I'd give them something to really worry about, by inviting a couple hundred of my closest camoflauge-wearing, open-carry, friends over for a little target practice on your backyard range for the Fourth.
Hoist up the biggest American flag you can find and be sure they are all well behaved and no alcohol is served, that way you are perfectly legal. If any of them own large O.D. vehicles, (tracked are best for effect) be sure they are also prominently displayed (legally parked of course). If the nosy neighbors drive by, be sure that they are loudly, and warmly invited to join in each time they drive by. Using the largest, and angriest looking member of the group is always a good way to do this.
Before all of this, be sure to call the local authorities to let them know you plan a large bash, and get the necessary exemptions for parking, amount of people allowed at your home, etc.
See if there are any more calls to the local authorities after that.
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