While not an answer to your question specifically, I want to note that Constantinople's population varied over the course of Byzantine history (330-1453).

The time period that likely saw the closest comparison to the slum-like tenements of Ancient Rome was in the late 5th-early 6th century. With anywhere from 40-80 thousans crammed into the eastern half of the peninsula, this was a time of social instability and during the reigns of Anastasius (491-518) and Justinian (527-65) major riots tore through the city. These troubles were fuelled by the surplus of poor, unemployed men looking for something to do and were often recruited by the circus factions (the Blues and Greens) who made a habit of causing problems in the streets and tried influencing/overthrowing Roman Emperors on several occasions. Famously, the Nika riots of 532 nearly overthrew Justinian and only at the cost of hundreds possibly thousands of lives was order restored.

The overcrowded conditions likely aided the Bubonic Plague that hit the city in 541 and may have killed anywhere from 1/4 to 1/2 the city's population. Add to that the devastating 7th century wars with the Persians and Arabs which caused the grain dole to be cut, Colin McEvedy estimates the population of Constantinople falling to between 15 and 20 thousand people. So if anything, the city would have been dealing with abandoned buildings rather than overcrowded ones.

Starting in the late 8th century, Constantinople's population began rising (the Bubonic Plague stopped hitting Constantinople after 750) likely peaking in the 11th century around 30,000 and staying steady there until the sack of 1204. During this period we begin to hear hints of the social unrest and overcrowding issues of earlier times notably during the later reign of Manuel I (r. 1143-80) and Andronicus I (r. 1183-85) who took power thanks to the mob and allegedly tore Andronicus apart when he was overthrown.

The crusader sack of 1204 caused the population to fall but after Byzantine reconquest in 1261 it seems to have rose to somewhere between 20 and 30,000 people when it fell to the Turks in 1453. The city was in much poorer shape due to the sack but the inhabitants, consistent with the late medieval trend toward urbanism, continued to move into the city.

Sources:

Colin McEvedy, Cities of the Classical World

Warren Treadgold, The History of the Byzantine State and Society