Justinian II had the Latin name Flavius Iustinianus Augustus and was the last Byzantine emperor from the Heraclian Dynasty, but his successor Philippikos Bardanes did not have a Latin name. Why didn't the Byzantine emperors going forward, such as the last one, Constantine XI, have Latin names?
From the Wikipedia entry on the Bizantine Empire (emphasis mine):
The emperor Diocletian (reigned 284–305) sought to renew the authority of Latin, making it the official language of the Roman administration also in the East, and the Greek expression ἡ κρατοῦσα διάλεκτος (hē kratousa dialektos) attests to the status of Latin as "the language of power." The scholar Libanius (4th century) regarded Latin as causing a decline in the quality of Greek rhetoric as the study of Latin became necessary for those who wanted to occupy public offices. In the early 5th century, Greek gained equal status with Latin as official language in the East and emperors gradually began to legislate in Greek rather than Latin starting with the reign of Leo I the Thracian in the 460s. The last Eastern emperor to stress the importance of Latin was Justinian I (reigned AD 527-565), whose Corpus Juris Civilis was written almost entirely in Latin. He may also have been the last native Latin-speaking emperor.
The use of Latin as the language of administration persisted until adoption of Greek as the sole official language by Heraclius in the 7th century. Scholarly Latin would rapidly fall into disuse among the educated classes although the language would continue to be at least a ceremonial part of the Empire's culture for some time.
Regarding Heraclius, the respective Wikipedia entry states:
One of the most important legacies of Heraclius was changing the official language of the Empire from Latin to Greek in 620.
Page 311 of John Julius Norwich’s Byzantium: The Early Centuries states (copied from here):
“If Justinian had been the last of the truly Roman Emperors, it was Heraclius who dealt the old Roman tradition its death-blow. Until his [Heraclius’] day, Latin was still regularly used by the civil service and even by the army — despite the fact that it was incomprehensible to the overwhelming majority of his subjects. At a moment when efficiency of communications was of paramount importance [i.e., during the Arab attacks], such a state of affairs was clearly ridiculous; and it was Heraclius who decreed that Greek, for long the language of the people and the Church, should henceforth be the official language of the Empire. Within a generation, even among the educated classes, Latin became virtually extinct. Finally, by way of marking the end of the old Empire and setting the seal on the new, he abolished the ancient Roman titles of imperial dignity. Heretofore, like his predecessors, he had been formally hailed as Imperator Caesar and Augustus; all these were now replaced by the old Greek word for ‘King’, Basileus — which as to remain the official title for as long as the Empire lasted.”
The truth however is that some of Heraclius's successors (but not all of them) did use a Latin title too. As you mention, it was Justinian II, 80 years later, the last emperor to use a Latin name (see list of emperors here).
So, if anything of the above is true, the disuse of Latin as an imperial title/name was just part of the cultural dimension of the continued historical trend of discord between the "East" and "West" of the (former) Roman Empire, divisions which ultimately led to the Great Schism.