For reference, the inscription of the aforementioned medal was "Orderi di Danilo", an Italian (?) translation of the native name for the medal. I doubt there are many Montenegro experts here, but any general information would help too.
I can't speak in much detail for the historical side of this, but I am prepared to debate this book to the death. My interpretation of that scene was always that the medal was fake (a part of Gatsby's reinvention of his identity following the war) and Gatsby had failed to do proper research and that's why the inscription was wrong. It is of course entirely possible that the error was on Fitzgerald's part, not Gatsby's. There wasn't exactly an Internet to quickly look up details like this back then, and likely very few people Fitzgerald could have met who would have even seen the medal before. However, I find the mistake very in tune with the way Fitzgerald wrote Gatsby's character.
And the inscription in the book absolutely is wrong. Orderi appears to be the word for Order in several languages (Google suggests Corsican), but as you pointed out, a Montenegrin medal would be expected to be inscribed in Montenegrin. This particular award was given to both Montenegrins and foreigners who performed services for the crown.
This is one example of the medal in question, a first class Order dated 1910
The zoomed in view very clearly shows that it is indeed inscribed in Montenegrin, which uses the Cyrillic script, not the Latin alphabet. I freely admit that I can't read a bit of it, although I can say that it doesn't appear to read Орден Књаза Данила I, (Orden Knjaza Danila I), the actual name of the medal.