I know that Istanbul used to be called Byzos (or Byzantium), and that Constantine renamed it in his honor. But the city had long since dropped the original name by the time of the fall of the Western Roman Empire. The city wasn't renamed again until Mehmed II, yet the people called themselves Romans. So why do we call it the Byzantine Empire and not the Constantinopolitinian Empire or Eastern Roman Empire?
Byzantium is referred to as such in order to distinguish it from the Roman Empire, as the two came to be highly different from one another. Byzantium was not just a 'successor state' to the Roman Empire but a largely different entity in terms of its institutions of state, society, culture, economy, military and geography. While Byzantines identified as Roman and felt much affinity to old Rome, their differences with the classical Empire has caused modern historiography to separate the two for study. Because of this distinction, you won't find any academic who treats both Roman and Byzantine history as the same field, and also Rome's founding to Byzantium's demise covers something like 2000 years.
And I'll agree that Constantinopolitan Empire would work as nomenclature for the Byzantine Empire, and is perhaps more apt as Constantinople came to always be at the centre of Byzantium's existence whereas Rome lost its importance to their Empire. But it's important to note that Byzantium's roots don't begin with the founding of Constantinople in 330 and is perhaps why we don't call it that. Also 'Constantinopolitan' is a bit of mouthful and when Hieronymus Wolf came along in the 16th century with his Corpus Historiae Byzantinae, the name caught on partly because of the contemporary appreciation of the classical era.
As an aside, Mehmed II didn't rename Constantinople. It kept that name, or the derivation of 'Kostantiniyye,' until the advent of the Turkish republic in 1923 when it was renamed Istanbul. But saying this, a Castilian diplomat said something interesting upon visiting Constantinople in October 1403:
The Greeks do not know the city by the name of Constantinople, as we are wont to call it, but name it commonly Estambol.
Ruy Gonzalez Clavijo, Embassy to Tamerlane, translated by G. le Strange, 89-90.
"Byzantine" and "Byzantion" were still used in the Empire long Constantine founded the city. While they existed in no official capacity, they were frequently used as an informal term for Constantinople and its citizens. Anna Komnene, for instance, uses it in the Alexiad in such a fashion. It is not a perfect term but it does have some form of precedent.
Sources: The Alexiad, Norwich (Yes, I know, somewhat suspect as a source), and I think Herrin briefly mentions it too (though I may be wrong on that count).