In more ways than one. Granted Silver could have snuck in since that idiot Gatekeeper had already carved his decision in stone and couldn't be bothered doing anything to change it, but this way it'll make the Duke house desperate, more inclined to side with Leo and help out, and make them owe Leo much more. Since Prince Leo 'went out of his way' to call Silver to come help and Silver was rudely turned away at the gate it's a huge insult to both Leo and Silver. Granted there was no advance notice from Leo or the Guild about Silver showing up, but calling Silver a fake and refusing to even examine his guild card was beyond the pale. Since Silver was rudely turned away it's implying the Duke house feels like it's superior to both the prince and Silver, that it doesn't need their help, and aren't afraid of the repercussions. Essentially it's like making an enemy of both the prince and Silver which means the Duke house is in really hot water politically. Also, since unable to handle monsters themselves that would mean hiring adventurers. However, if an SS class adventurer like Silver is treated so badly it means the lower ranked ones would be treated even worse so once word gets out the Duke house won't be able to hire any adventurers for anything.