The monarch is head of state The King also chairs monthly meetings of the Privy Council, to approve Orders in Council; he receives incoming and outgoing ambassadors; he makes a host of other appointments, such as the senior judges, but in all this he acts on the advice of the government.
Possession of an unregistered firearm is also covered by federal laws and can also be charged as a federal offense. 26 U.S.C. § 5861(d) provides that it is a crime for a person to receive or possess a firearm not registered to them in the National Firearms Registration and Transfer Record.
Under the legal doctrine of “sovereign immunity,” Charles is exempt from criminal and civil proceedings as the head of state. But the King's immunity extends beyond his public duties to his conduct on privately-owned assets, estates, and businesses.
While carrying firearms is heavily regulated for civilians, members of the Royal Family may travel with licensed firearms for personal protection or ceremonial purposes. Certain beaches in the UK are privately owned by the Crown Estate and accessible only to members of the royal family and their guests.
Sports (target, practical shooting etc) or hunting in England, Scotland and Wales are the primary reasons, but others like needing to put animals down humanly exist. In Northern Ireland in very rare cases you can add self defense. Sport shooting is not limited to the upper classes.
This is a long-standing constitutional convention that the monarch can do no wrong. This means that the King cannot be arrested, charged, or tried for any criminal offence or put under a civil investigation.
The King Kong franchise has traditionally followed a classic tale without the ape using a gun. It's possible Kong could use a similar weapon in the Monsterverse depending on what happens in Godzilla x Kong: The New Empire.
It is difficult to estimate how many illegal firearms are held in the UK and estimates have varied from several hundred thousand to over a million. The illicit firearms market could supply terrorist acquisition, however, this would rely on the use of a criminal contact.
Concealed or open carry of any weapon is generally prohibited in Great Britain (i.e. England, Wales, and Scotland), the Prevention of Crime Act 1953 prohibiting this in a public place. Permission exists only with lawful authority or reasonable excuse.