The idea that the Roman Empire ceased to exist after the sack of Constantinople in 1204 is a perfectly legitimate idea in historiography. But it's not that simple. There are a dozen possible "end dates" for the Roman Empire and none are necessarily incorrect.

In History, we like to label Empires and Kingdoms of the past as if they were modern nation States. This helps major political events in the state's history to stand out the way they do in modern states. For exsample the modern day Russian Federation is not simply the Soviet Union minus some lost terrorties but an entirely different state. But the concept of the nation state did not exist until at least the late 17th century and that kind of political distinction did not exist in Antiquity or the middle ages. A person living in Baghdad in 747AD would not change the way they referred to their state despite historians marking a clear distinction between the Umayyad and Abbasid Caliphates.

The Imperium Rōmānum, or in Greek Basileía tôn Rhōmaíōn is often translated as "Roman Empire". There's nothing inherently wrong with this translation so long as you understand that both "Roman" and "Empire" meant different things to different people at different times. To the Greek speaking Rhōmaîoi of Constantinople in 1204 it would of meant this.

Rhōmaîoi or "Roman" was a national and culteral identity much like "French" or "Russian". It had almost nothing to do with the City of Rome which at one point in the 6th century had fallen to a small town of barely 25,000 people. The national identity of the "Roman" was based on a distinction from "Non-Romans". Romans always had great pride in the laws and Constitution and this remained the defining trait of being a "Roman" long after citizenship became near universal. Until around the time of the 4th crusade the terms "Hellenic" or "Greek" were seen as offensive. After the Sack of Constantinople the remaining Roman elite such as the Nicean emperors looked to ancient Greek cultural heritage and orthodox Christianity to strengthen their claim to being Roman. This led to Romanness becoming increasingly associated with people who were ethno-culturally Greek. Rhōmaîoi survived the fall of Constantinople shifting to a cultural identity based on legends of the fall and the loss of their Christian homelands and Constantinople itself. This Roman identity persisted until the establishment of modern Greece in the 19th century and the Turkish "Rum" meaning Roman is still used to refer to native Orthodox Christians in Anatolia today.

"Imperator" as many well know is a term that originally roughly translates to "Commander" though by the early Empire the act of being proclaimed Imperator became synonymous with the act of Succession. By the 6th century, In the view of medieval Christians, the Roman Empire was indivisible and its emperor held some authority over Christians who did not live within the formal borders of the empire. As such there could only ever be one emperor (Basileus) which after the coronation of Charlemagne led to the problem of the two emperors - The ideological contradiction between the religious theory that the emperor has universal rule over christendom and the reality that there were two or more people at any one time who claimed to be an emperor. After 1204 this got significantly more complicated as the Latin emperors also claimed the title of emperor, as did the rulers of Nicea and Trebizond. This period is more akin to a multi-sided civil war than an occupation and is honestly quite a mess. But while there was more than one person claiming the rightful title of emperor, This was hardly the first time that had happened. After the the 1261 conconquest the new Palaiologos dynasty largely abandoned any claim to universal rule in order to maintain better relations with the west with the goal of eventual reunification of the churches. From this point on the title of Emperor no longer meant what it once did as the Roman Empire essentially became a vassal of the Ottomans. In this respect you could argue that the Roman state was no longer an empire in either the modern or medieval sense. However the Ottomans DID consider the Roman empire to have been an empire right up until 1453 in order to claim the title Kayser-i Rûm (Ceasar of Rome) for themselves by right of conquest. The two emperors dispute continued with the HRE and Ottomans and would not be solved until both recognised their respective titles in 1606.

In conclusion:

From a cultural standpoint the Roman identity become more closely associated with Hellenism after 1204 but "Roman" remained the dominant identity of ethnic Greeks until the 19th century.

From a technical standpoint the definition of "empire" was stretched post 1204 in both the modern and medieval definition. Rulers of Nicea, Trebizond and the eventual Palaiologos restoration gave up their claim to universal rule, became effective vassals of the Ottomans after the 1330s and no longer ruled a particularly multi ethnic or large terrority.

From a legal standpoint the title of emperor was transferred directly from Constantine XI to Mehmed to Conquerer by right of conquest in 1453 and was recognised as such by the HRE and the Habsburg monarchies after 1606.

TLDR: It depends