Is anyone aware of how one might refer to this sort of title ?
I would like to submit this question:
What would your <insert word I don't know> be if you were a monarch? (e.g. Frank the Foolish, or Hillary the Hot, etc.)
A descriptive word or phrase that customarily accompanies a person's name is traditionally called an epithet. The classic examples are from Homer. To give the most famous illustration, in the Iliad, Achilles is usually "swift-footed Achilles" or some other version of an adjective for swift combined with his name. For example:
In answer to him spoke swift-footed Achilles, "Take heart, and speak out whatever oracle you know…"
In answer to him spoke swift-footed brilliant Achilles, "Most glorious son of Atreus, most covetous of all, how shall the great-hearted Achaeans give you a prize?"
But he in his wrath sat beside his swift-faring ships, the Zeus-sprung son of Peleus, swift-footed Achilles.
"Brilliant" and "son of Peleus" are two more of Achilles' epithets. The word "epithet" is actually Greek for "adjective".
However, I think that most fluent English speakers today would not understand you if you asked them "What would your epithet be if you were a monarch?" In common usage, the word epithet has gradually become specialized for pejorative names. For example, "racial epithets" means racial slurs. Consequently, today the honorary or descriptive sense occurs mainly in scholarly writing.
So, today there is some pressure to favor other synonyms, such as sobriquet, appellation, byname, cognomen, moniker, or even nickname. Some people object that sobriquet should be limited strictly to substitutes for a person's name rather than descriptive phrases adjoined to a person's name, like "The Bard" for William Shakespeare. I don't think that prior usage was ever really so strict, though. The OED* does not suggest that any of these are or are not reserved for customary descriptions adjoined to monarchs' names. Nickname is traditional and well-established for monarchs; see, for example, this Wikipedia article. However, most people don't associate a monarch's epithet with the word nickname, and would misunderstand if you asked them "What would your nickname be if you were a monarch?"
As often happens in English, you are faced with an unpleasant choice: use the traditional word, well established by centuries of continuous precedent in writing and in speech, and be misunderstood; or use a clearer word and be criticized—often wrongly!—for ignorance or sloppy usage. I would probably say sobriquet or appellation and accept the risk of slings and arrows. You can always switch to another synonym if the other person complains.
* The Oxford English Dictionary defines sobriquet as "An epithet, a nickname."
The general structure is as follows: a proper noun (Alexander) modified by a postpositive adjective (Great) which is preceded by the definite article (the).
As for whether there are specific terms for this kind of construction, I've found a similar question on ELU, and there are three answers to it.
Quoting the most highly voted answer:
This particular usage itself - an adjective following and considered part of a name - is most commonly referred to in English as an epithet.
Quoting John Lawler the Reputable Linguist:
The adjectival constructions are just normal generic uses of adjectives as appositives.
This kind of title is an "honorific" or "style". Unfortunately, your question mixes up the epithet as described here by others (which, by the way, is usually a negative connotation - "a racial epithet") "Frank the Foolish" as what a monarch might be called instead of what a monarch might actually be called, such as "Her Majesty the Queen". In English, honorifics are almost always at the front of a name, like Dr. or Rev., whereas a style might be "Your highness" or "Your Excellency".