I would like to recommend Anthony Kaldellis' excellent 2015 book The Byzantine Republic: People and Power in New Rome, which opposes the traditional narrative of the Byzantine Empire (as a theocratic autocracy that had evolved beyond recognition from its Roman republican origins, etc.) with a revisionist account that quite forcefully explores the implications of what we already accept about Byzantine history without much questioning: viz. that the Byzantines routinely deposed and replaced reigning emperors, for reasons that are largely unintelligible in the received divine right/caesaropapism framework, and without apparent damage to the greater cultural and normative institutions that constituted the empire and its culture.

Kaldellis' account is of the medieval Roman empire as a fundamentally republican entity (in the ancient Roman sense, or something very closely resembling it), and is, as you may guess, deeply revisionist; although he probably takes his thesis a little bit too far, his work is really a refreshing antidote to the general paradigm that the Roman state inexorably mutated into ever-deeper theocratic despotism as the centuries rolled by.

Indeed, the holders of Imperial Office did continue to use the pretense of Republican institutions even after the fall of the Western Empire. As you mentioned Constantinople also had it's own Senatorial class, who elected Consuls and served as public officials. The Constantinopolitan Senate like it's Western counterpart also held considerable political influence for quite some time, with some Senatorial families taking part in the Nika Revolt to attempt to overthrow Justinian.

The Nika Revolt coincides (maybe not coincidentally) with the lapse of the Consular office, with Anicius Faustus Albinus Basilius being the last independently elected Consul. Justinian also got rid of Consular dating and replaced it with Regnal dating (Dating following the time of an Emperors reign).

Following Justinian's successor, Justin II the title of Consulship would be transformed into a part of the Emperor's coronation ceremony. The Emperor would take the title of and be "elected" Consul as part of the rites of proclamation. Due to this the office of Consul was exclusively held by the Emperor.

Leo the Wise got rid of the honorary rite of Consul during his reign, as by his time the term Consul (Greek: ὕπατος) was reserved for lower dignitaries or the lower officials of Constantinople.

Despite the title of Consul dissapearing (or transforming it's meaning into something significantly less honorable) the Senate would continue. But it's powers would also fade with the turbulent time of the Byzantine Dark Ages (7th-8th Century), in which many Emperors absorbed many of the Senate's legal powers into their own office. Granted the Senate still held consierable prestige, but it had faded as a public institution and Senators became more like dignitaries or advisors for the Emperor.

By the Komnenian era and beyond it becomes increasingly difficult to talk of a "Senate" as while the term was used, it's meaning could be more accurately interpreted as referring to the local government of Constantinople.

So in conclusion, yes the Emperors did still base much of their legal authority and legitimacy on the ancient Republican institutions. However the power and importance of these institutions would greatly transform over time, and some like the Consulship would dissapear entirely.