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Historically, these kinds of duties would likely fall to the Vizier or Chancellor or Prime Minister.

Looks like you'll be wanting to come with some title names other than, e.g. "commander of war". Given your constraints, Chancellor might work well. Or alternatively, Viceroy.

Have you considered such titles as:

• Plenipotentiary


• Legate


• Deputy


• Arbitrator


• Consul


• Proconsul


• Nuncio


• Internuncio


• Attaché


• Premier


• Factotum


• Procurator


• Proctor

Shogun or Regent

The problem with this arrangement is that such position involves an enormous amount of power, eclipsing the power of Emperor himself. Anyone who has an authority to negotiate on behalf of an empire has special powers. Anyone who is the primary authority for all negotiations, both domestic and international becomes the primary authority of this empire. Why talk to the Emperor if you can talk to his master of negotiations?

In real life, such authority would become a "power behind the throne", with real Emperor withdrawn from politics and decision making - either voluntary, or deliberately restricted to a ceremonial role.

P.S. This is different from a "trusted advisor" scenario, in which Emperor is calling the shots, and dispatches his advisor to conduct negotiations on case-by-case basis. "Trusted advisor" has little permanent powers except having his master's trust.

In the Holy Roman Empire, the Prince Electors, the highest after the emperor, held the titles: Arch-Steward (or Arch-Seneschal), Arch-Chamberlain, Arch-Marshal, and Arch-Butler. These all originate in the offices of a royal household, but all save the Butler have since lost most of their connotations with domesticity and could fill in for a title of a sort of prime minister.

The Steward was originally head of all household staff, but this consummate manager could, over time, start to oversee more and more of the Emperor’s lieutenants in the realms of economy and statecraft.

The Chamberlain was in charge of the Emperor’s wardrobe and bedchamber, which seems trivial in modern times, but this meant very close access to the Emperor’s person. Only someone trusted should have access to where you sleep! In the Byzantine Empire, the keeper of the wardrobe (known as the Vestiarion) eventually came to control the entire treasury in a great example of scope creep.

The Marshal was in charge of the Emperor’s horses, and from there it’s not a stretch to oversee the cavalry and soon the whole military.

The Butler or ‘bottler’ was in charge of wine specifically and the kitchen more generally. This one seems especially silly to modern audiences, but could actually be kind of cool if you take the time to explain it. Being in charge of food and drink means you’re the Emperor’s first line of defense against poisoning, which is also why a Cup Bearer was so important in the Middle Ages. You could show how the Butler came to control security of the Emperor’s person more generally, and from there to ever more vast responsibilities.

Your official who represents the concerns of the citizenry can be an ombudsman.

https://en.wikipedia.org/wiki/Ombudsman

The typical duties of an ombudsman are to investigate complaints and
attempt to resolve them, usually through recommendations (binding or
not) or mediation... At the national level, most ombudsmen have a wide
mandate to deal with the entire public sector, and sometimes also
elements of the private sector (for example, contracted service
providers). In some cases, there is a more restricted mandate, for
example with particular sectors of society.

And I love to say "ombudsman". Try it!

In the modern age, I doubt there is a single individual who would handle both foreign and domestic diplomacy, however what you are really describing is a person who can act on the King's behalf in the King's absence.

The word for such a person is... "Proxy"

• Councillors, especially Privy Councillors.
  
  These are persons who regularly confer with the ruler in confidence, and who can be expected to both speak the ruler's mind and carry messages back directly to the ruler.


• Agents, especially Agentes in Rebus.
  
  These are persons who have been designated by the ruler to "look into matters" -- whatever matters the ruler thinks need looking into.


• Princes and Princesses.
  
  Relatives of the ruler. When they speak, they speak as members of the ruling dynasty, and the ruler couldn't repudiate their promises without dishonoring his own blood.


• Dukes, or Counts.
  
  If the Empire has an Emperor, it is probably not a democracy. Persons who are nobles in their own right can make weighty statements.

When Cardinal Richelieu was "first minister" or "chief minister" of France, he delegated specific duties (of the sort described in the original post) to intendants.

Envoys, consuls, and ambassadors are also possibilities. In the Roman Republic, the two consuls were normally the highest officials. Ambassadors plenipotentiary are theoretically the most trusted of ambassadors.

In Lois Bujold's Vorkosigan series, "Auditors" are trouble-shooters who "speak with the voice of the Emperor". The books Memory, Komarr, and Diplomatic Immunity feature Miles Vorkosigan as an Auditor.

As you said, 'Diplomat' is good, but it's kind of a catch-all for anyone who does this type of work. There are apparently ranks within the general designation of Diplomat though, including Ambassador, Minister, High Commissioner, Counsellor and Envoy. (found on wikipedia here)

If you're looking for something a little less real world I'd suggest Negotiator or Peacemaker.

Imperial China had the title of "Grand Secretary". They replaced the earlier position of chancellor. Basically a Secretaries job was to screen which documents/papers from the different ministries where sent to the Empreror. This gave them tremendous power since they could selectively pick which documents to send to the Emperor or draft their own.

You could have a role very similar to this and call them something like "Grand Chancellor" you could have them be a position above the rest of the kinds cabinet who's job is to delegate which domestic issues have to be solved personally and which he can hand down to one of his own official. Think of it as a bureaucracy that's above the rest of the bureaucracy.

I'd also suggest giving him Powys that let him "borrow" materials from the other cabinet members. Ex: a riot in the slum so he borrows a company from the guard.