Fascinating question, not least because a moment's thought suggests that the type of staples you're describing would be pretty useless for any stationery-related purposes - gold is a soft metal and encrustations of "precious stones" would be an obstacle to any attempt to neatly collate paper.
A few checks reveal that this story has been pinging around online since the turn of the millennium on sites such as Mental Floss, almost always accompanied by the rather worrying prefix "legend has it". But in fact the story does antedate the internet; the earliest version I have found - which is, importantly, stripped of all mention of royal insignia, gold staples and precious stones - actually appeared in Investor's Reader, published by Merrill Lynch, in 1962. The reference appears in an article devoted to a stapler company, Swingline Inc. (formerly the Parrot Speed Fastener Co.), of Queens, NY, which noted that founder and chairman "Jack [Linsky] showed NYSE prexy [president] Keith Funston the original stapler developed by Louis XV and its streamlined replacement..." The piece references an illustration of the device, an example of which Linsky apparently owned, but this unfortunately cannot be viewed online [EDIT: I have now obtained a copy of this image – see link at the foot of this thread]. The "streamlined replacement" would have been one of Linsky's designs.
All this is interesting not least because Jack Linsky was the man who invented the modern stapler, creating the adhesive that is used to fasten lines of staples together so they can be loaded into a magazine. He emerges from histories of the staple as a man who took a serious interest in the history of the device, and I think it's almost certain that the story of Louis XV's "stapler" was first disseminated by him.
The reality, though, was that nothing created in this early period really matches the modern description of a stapler, and the detail that Louis's staples were jewel-encrusted and made of gold does not appear until the very end of the 20th century. Moreover, Linsky plainly had good reason to re-label early paper-fastening devices – and especially devices produced under the patronage of a royal court – as "staplers" when they were really nothing of the sort, since this dignified his own business with a long and royal history.
From the scant information available, it appears that there are two broad possibilities. One, mentioned by an Italian industrial design website, is that the machine was actually "a tool that uses decorated pins, with which the king imprints the royal seal". The other is that the "staples" used at Louis XV's court were a type of paper fastener, used to secure papers in place of the sealing wax or tape or string then used. This would mean that the device described by Linsky was in effect some sort of binding machine, perhaps more accurately imagined as a distant precursor of the modern saddle-stitch binder, which would have been used to pierce the papers and insert a metal fastening and bend or twist it so it secured the papers.
The practicality of such a device would have been dubious, as it was not until the development of steel wire in the second half of the 19th century that a wire with the correct combination of softness and durability to be useful in fastening and binding papers was developed. This suggests that the Italian description is probably more accurate – though, if so, we'd be looking at something not unlike a Braille embosser that functioned in the same way as a stapler (insert paper, press down, use metal spikes to make holes in paper) rather than a device having the same purpose as a stapler. That is, it would not be any sort of paper fastener.
Unfortunately, despite the investment of some time and effort, I have not yet been able to trace any mention of the original machine referenced by Linsky in more general sources relating to the history of paper fastening technology or the court of Louis XV – very probably because such sources refer to the device by its unknown proper name. I have a query currently pending at the Library of Congress in an attempt to secure a copy of the illustration in Investor's Reader and when that gets answered, I'll update this response.
In the meantime: modern staples of the sort that are dispensed by machine date only to the 1840s, with the invention of devices such as Samuel Slocum's "Machine for sticking pins into papers" (1841) and Lipman's Improved Eyelet Machine (c.1868), a form of modified rivet for use by clothing manufacturers. The word "stapler" dates only to the 1880s; it first appeared in The American Bookseller of 2 June 1884 to describe a new type of "novelty paper fastener".
And the humble staple was a successful enough innovation to make Linsky rich. A New York Times profile of him notes that
"He and his wife, Belle, were philanthropists and art collectors who once owned one of the largest collections of Fabergé eggs in America. Jack died in 1980, and in 1982 Belle donated a collection of the couple’s European art, then worth $60 million, to the Metropolitan Museum of Art."
The Linsky Art Collection – which, perhaps not coincidentally, is quite heavily focused on pieces created during the reign of Louis XV – is now an important feature of the Met. It seems to me quite possible that the device used in France in Louis XV's time, and which was once part of the Linsky collection, still survives somewhere as well – although, having reviewed the Met's catalogue, it appears it was not one of the items transferred to that museum by Belle Linsky.
More when I have it. But for now I think it's reasonably safe to conclude that the basic story originated with Linsky and that the accoutrements - not least the jewel-encrusted staples - are just inventions added by imaginative writers to make the story more interesting around 2000. So, the device you are interested in was almost certainly not a stapler and did not dispense gold staples, jewel-encrusted staples, staples imprinted with the royal insignia, or any combination of the three.
Sources
Phyllis Korkki, "Staples, the attachment that still makes noise," New York Times, 23 March 2013
Philippe de Montebello et al, The Jack and Belle Linsky Collection in the Metropolitan Museum of Art (1984)