Would Prime Minister Boris Be Unconstitutional?

I have noticed something when you look at the list of Government ministers. If you look at the Whips' Office, you come across:

You'll notice that several of the Whips have the title Rt. Hon. in front of them. This is short of Right Honourable and indicates that they are a member of the Privy Council, having sworn:

You do swear by Almighty God to be a true and faithful Servant unto The Queen's Majesty as one of Her Majesty's Privy Council. You will not know or understand of any manner of thing to be attempted, done or spoken against Her Majesty's Person, Honour, Crown or Dignity Royal, but you will lett and withstand the same to the uttermost of your power, and either cause it to be revealed to Her Majesty Herself, or to such of Her Privy Council as shall advertise Her Majesty of the same. You will in all things to be moved, treated and debated in Council, faithfully and truly declare your Mind and Opinion, according to your Heart and Conscience; and will keep secret all matters committed and revealed unto you, or that shall be treated of secretly in Council. And if any of the said Treaties or Counsels shall touch any of the Counsellors you will not reveal it unto him but will keep the same until such time as, by the consent of Her Majesty or of the Council, Publication shall be made thereof. You will to your uttermost bear Faith and Allegiance to the Queen's Majesty; and will assist and defend all civil and temporal Jurisdictions, Pre-eminences, and Authorities, granted to Her Majesty and annexed to the Crown by Acts of Parliament, or otherwise, against all Foreign Princes, Persons, Prelates, States, or Potentates. And generally in all things you will do as a faithful and true Servant ought to do to Her Majesty so help you God

When you look at the list again, you note that the senior Whips are all Privy Councillors - except for Deputy Chief Whip and Comptroller of Her Majesty's Household, Greg Hands.

I have wondered about this, but note that Hands - like the Mayor of London, Boris Johnson - was born in New York City, and therefore is an American citizen.

If we look at the American Constitution, then Article 1, Section 9, states inter alia:

No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State

This has got me thinking - would becoming a Privy Councillor count as accepting an "Emolument, Office, or Title" from a foreign State? Is this why Hands is not a Privy Councillor? And, if so, would this rule out Johnson taking any position - such as a member of the Cabinet - that would involve becoming a Privy Councillor?