Why a golden bullet?

Because this is the arrangement he has made with his 'employers'. The use of the gold bullet was the way he signatured his assassination of the general. When the reports of the killing came back to the employers, the detail of the gold bullet would confirm that Tate had been the man squeezing the trigger. Upon reciept of this information they paid him his gold dollars for services rendered. It's another very clever piece of plotting, by which the character's methodology is revealed stage by stage.