Our current party is at level 16, and has been at level 14+ for years (We play weekly, but do not give out exp, we only level up when the group decides to, the campaign started back in 2E). It's set in the Forgotten Realms, but with less magic items and mages, and not many if any epic characters. So, the characters are near the top of the "food chain" level-wise.
More important than that however is that they have a very, very good reputation in their country. They battled countless foes of the realm, spoiled plots inside and outside the country, handled diplomatic meetings and put down rebellions. The earned the gratitude of the ruler, and rose in status among the ruling churches, earning titles.
While I am not a "simulation tops everything!" DM, verisimilitude means that unless the group goes directly against the ruler or a church, they can basically do whatever they want. If an NPC is rude to them, especially a foreigner, they could kill said foreigner, and the matter would be handled either by hushing it up, arranging an "it was an accident", or by posthumously condemning the victim. Or by simply ignoring the incident. Or by acepting their deed as justice - after all, they are part of the government. That's in their country, of course. But everyone knows that - and there really are not many who would insult them.
Outside their country, it depends on the situation. When travelling on a diplomatic mission, they have the power of their country behind them (unless they do something really stupid, like starting a war), and if someone is rude or insulting to them, it's usually handled by a duel.
So, if they were in that country from the OP, and the arbiter came to them, and acted in the way described, they'd tell him, quite haughtily, to not mess with them, and to suffer the consequences of attacking not just a representant of the Empire, but also a priestess of one of the ruling churches. If they felt insulted, there'd be a duel challenge, and if refused, most likely the "honorless cur" would be dealt with. And the king of the realm would apologise for having employed a "temporarily insane" idiot. Behind the scenes, there'd be all kind of maneuvers, and some repercussions, but the king would not go to war over this.
But then, the arbiter in my campaign would not have tried to bully them, would probably have been all nice and polite, and gotten the baby without any trouble just for mentioning that he was his father - the group generally respects lawful authorities unless there's something fishy - like said authority refusing to answer basic questions.
I've played the standard campaigns before, where PCs rise in levels, and can flatten mountains, yet are still treated like some dinky mercenaries by nobles. It mostly did not really work out that well, and felt contrived. Once I started adjusting the status of the PCs, and integrating them in the power structure, it went far more smoothly. Power now comes with reputation, respect, status - and responsibility.
The carrot of rewarding PCs with status worked far better than the stick of using high-level NPCs to beat them down. More fun for everyone involved.
And in areas like the City of Brass, the PCs still are pushed around, because they have no weight there. But they know that they can change this.