If the sheriff is the closest and easiest to get to, I'd at least give him a try.  Explain that its in .223, so even supressed, its still as loud as a 22LR.  You can also stress the importance of protecting your hearing and that you are trying to be considerate of your neighbors by keeping the noise down.

If he can't be swayed, you can also try the head of the state police, a DA or prosecutor (state or local) who has jurisdiction over your place of residence.  If they all fail, you can go back to the sheriff and present him with a choice - [i]this is a last resort[/i].  He either gives you the signoff as an individual or you will go corporate and buy one anyway, thus completely circumventing his authority.  That gives him a hard decision to make.  He can either grant you the signoff you really want (and in return, he gets to know what you're buying and have in your posession) or you can go corporate and he'll never know what you have (supressors, machineguns, AOW's, etc), how many, what caliber, etc.  Essentially, you can explain to him that you will legally get the supressor, with or without his consent, but if he consents, he will at least know whats in his jurisdiction.  Now granted, you may not really want or intend to go corporate, but you can use it as a bargaining tool.