Edit: I posted this comment in response to someone who said "there was no Byzantine Empire in 1247"—and either the mods or the author deleted it. So, that is why the first paragraph is awkwardly framed and perhaps rather oppositional.

I don't like this answer, because it sort of suggests a sort of civ-esque understanding of history wherein if the capital is seized, the Empire is defunct. Most historians that I am aware of—although I am not a Byzantinist—speak of this period as something between an empire in exile or an interregnum.

Various Byzantine families that established rival claims to the throne not withstanding, rulers continued to style themselves according to Byzantine culture and style. Looking through the wonderful catalogue of coinage digitized by Dumbarton Oaks, there isn't an abrupt shift in iconography from the period before 1204 and the period after. The Laskaris in Nicaea continue to use the iconography of Empire, so I think it would be inappropriate to call the Empire non existent during this period. Indeed, I believe in their correspondence the dynasty continued to identify themselves as the "Emperor of the Romans," the Emperors of Nicea being a post-historical construct.

I'd welcome someone who knows more about Byzantine documentary culture to add to this discussion.

I have no idea if the Byzantine Empire celebrated the two thousandth anniversary of the Empire, but at least from the evidence I've seen—what texts people wrote and consumed as well as how medieval cultures commemorated stuff—it would be unusual. While classical texts were alive and well in the Greek manuscripts of the Byzantine Empire, they aren't drawn on heavily by those writing history at the time. Authors of the thirteenth and fourteenth centuries tended not to draw on the classics as much as earlier Greek historians, such as Michael Psellos—who begins his Chronographia in the 10th century with Basil II—or Leo the Deacon, whose history covers two Byzantine Emperors of the eleventh century. In other words, the preferred historical playlist was of the contemporary, Hellenized past rather than what we would consider as early Roman.

In addition, celebratory events in the medieval world tend to revolve around important dates on the liturgical calendar, as well as what would have been considered important in the larger family memory: coronations, assumptions of office, notable deaths, etc.