That's a very interesting problem. Maybe you could try adding some dilute nitric acid little by little to slowly remove the carbonate ions (by conversion into unstable carbonic acid and then to carbon dioxide gas). Don't use HCl since chloride ions will be present to precipitate with your silver, which will be a problem. *edit* but you still risk making H2S by adding acid as Borek mentioned, so this isn't the best way to go about the problem.
If you don't absolutely have to use acid/base, I'd try a different approach. I would introduce Ca2+ ions into your solution to precipitate all of those carbonate ions first (maybe in the form of calcium nitrate if you have any or some other water-soluble calcium compounds). Then filter and go ahead to use your silver nitrate to isolate your sulfide ions.