Names becoming titles is definitely a thing, but perhaps more often than not what we take as a name was really a title. Caesar was a family name of sorts if I understand correctly. Despite seeing it explained I don't really understand the Roman classical naming system; both the Romans, who appear to have had a very limited supply of acceptable normal personal names, and the Greeks with a larger supply of them but no established general family name system I have noticed, were prone to giving notable people extra terms such as "Aristedes "The Just"' or what have you, and the familiar widespread system of someone going by an extended name saying what city or town they hailed from is a major thing too.
Personally I have enjoyed suggesting a TL where the Roman Dukes of Britain had their act together more, organized a hold-out of Romanized Britannic civilization and were around instead of Charlemagne (that is, Charles "the Great") to serve as the new western Emperor anointed by the Papacy--in that scenario I gather "Arthur" was not a given name but a Britannic honorific made up for a particular hero, meaning "Bear man" or something like that, and in the ATL this name is made up for a leading Dux of particular importance, either a predecessor of the Emperor or the same guy--and then the title of Augustus Arthur or some suitably Latinized form becomes established; "the Arthur" would be a term interchangeable with "Emperor" or "High King" should a language like OTL English evolve anywhere.
But if in fact Arthur were a common name equivalent to Bob or Joe, I don't suppose that would work so well.
Nor is there much precedent for this kind of thing in European Christendom; I don't think anyone ever expected the Norman dynasty would have its kings known as "The William" or the Plantagenets "The Richard" or what have you.
George is also terribly ambiguous in this context; while I don't think any names more recent than "Caesar," which did become the titles "Kaiser" and "Tsar" in German and Russian, became alternative titles to Emperor or king, it was and is very common for dynasties to keep naming the main heir a traditional king name, often the same one generation after generation, as Louis is typical of French kings, and as it happened the Hanoverian dynasty was stuck on "George" at this very time. A new king can of course take a new name and this where at least some "Georges" in the British succession come in but by and large they got this as their first name at birth.
So--there would be worse than the common danger of mixing up the American monarch with the Hanoverian succession should the Americans unaccountably (with all due respect to General and Dictator Washington) choose to honor their greatest (prior) leader with naming all kings "George." Much worse, because the whole sentiment for royalism in America, while I will not gainsay the author's assertion was popular enough to turn into a majority sentiment under the right conditions, was also a position on the political spectrum, one that more radical revolutionaries identified with some justice as conservative and perhaps, they would argue, unpatriotic, calling into question whether there ever should have been a Revolution in the first place and undermining the belief that national identity could take a different and in the revolutionary (radical anyway) view more rational form than identification with a monarch as a person and a royal line of succession. We know that at least some anti-monarchial democrats still exist despite being discredited by the extremism of certain recent events; even if they wind up criminalized by "lese majeste" legislation they will remain (fewer, but all the more diehard for being forced underground, and perhaps more liable to be politically active, if anonymously and criminally so).
So--the name "George" is unfortunate out of all possible choices because it could be read as a dogwhistle of hard core Tories who regret the whole Revolution and back an American monarchy as a substitute for the British one they never intended to break from or belatedly regret having been snookered into so doing. It gives ammunition and cover to American radical democrats who could indiscriminately so accuse all American monarchists of being secret Tories, and to attack monarchist policy as inherently unpatriotic.
If regnal names identical down the generations were the universal custom in Europe, it might be more of a dilemma, but in fact even nations like France and dynasties like their Bourbons allowed the predominant flow of "Louises" to be interrupted with the occasional Charles or Francois; the Danes and Swedes would mix it up a bit, even the Saxon royal house would mix up their George Augustuses with various permutations. The American thing to do would be to call a king a king, and just use their first name, whatever it happened to be.
Gilbert is a little weird to my ears but it definitely has a royal ring to it--especially if one's studies go back to the Dark Ages of course! I did not know until reading this thread that this was LaFayette's given name, I'd only ever seen him referred to as just "LaFayette" or even "Lafayette"--the former more correct version is flagged by my spell checker, the simplified run together version is not! I presume "La Fayette" would be most correct but my spell checker disapproves of that too, and apparently changing his allegiance from the King of France to become the American king would have the side effect of stripping him of claim to that French title he is known universally by OTL.
I agree that if Americans are going to give their new kings any distinctive unique new title, they will not borrow anyone's name (except insofar as Caesar or Augustus are actually names--the latter is particularly useless for being a common given name in various languages) but instead use titles borrowed from Latin or perhaps French, conceivably Greek though that would be quite dubious, possibly though quite unlikely Hebrew (those most likely to support it might also find it a bit sacrilegious, and also unwise in that IIRC the book of Chronicles has Samuel trying to dissuade the Israelites from adopting a king in the first place and warns them of calamities the new practice would bring on them--more political ammo for the republicans)! Probably not Anglo-Saxon because the vogue for finding roots of English speaking democracy in the pre-Conquest English tradition would not come for another century; maybe earlier in the Romantic period of literature, but it is still far too early for even that.
We'd just stick with "king," I think, with "royal" as an adjective, and leave it at that, maybe going for "emperor" from the get-go instead, or adding it later. No further weirdness is required and if a new king's given name is no good for a reign they will sparingly add a new one--probably not George, until either US relations with Britain are much better or the British royals have backed away from that name.
I don't see "Gilbert" becoming fashionable either though it could be the default name for someone named "Ichabod" or some such by their mother; by and large I think Americans would prefer to remember each king separately and distinctly with the drama of making up yet another new and exciting name for the heir becoming part of the tradition. Americans might pride themselves on never ever having to add a Roman numeral to any king's name, since each one is unique and distinct; assuming the former Marquise de La Fayette turns out to be a really good king, it might seem terribly presumptuous and asking for trouble to give any successor the same name by act of will. In fact Gilbert might become a common American name for everyone but the royal house.
Is it the plan at this point to not only have a monarchy, but a fixed dynastic succession as well? To some monarchists that is the whole point of having a monarchy, but others will point to the idea of an elective monarchy and hope each US king is chosen on merit--as Lafayette has been!