I am assuming by what you meant is that elements of the city directly or indirectly aided the Ottomans in capturing Constantinople but not that the Byzantines (and the Byzantine government) themselves completely orchestrated the Fall of Constantinople as in the case of 9/11 conspiracies.

You could consider the fact that many of the non-foreign captures of Constantinople occurred because factions of the city supported the sieging forces (often a emperor claimant) and aided them in capturing Constantinople by opening the gates or by sabotage of the defence effort. Therefore, historically, many sieges of Constantinople, albeit domestic ones at that, succeeded because of internal sabotage.

During the Siege of 1453, what is considered the turning point that gave the city to the Ottomans is what's infamously called the Kerkoporta Breach incident which actually does have flavors of potential conspiracy in it.

Doukas, a 15th century Byzantine historian from whom we give credit for much of the information on the twilight Byzantium, recounts the significance of the Kerkoporta, a small gate from where the Blachernae fortifications meet the Theodosian Walls.

During the later days of the siege, the Byzantines would send out sallying parties to attack Ottoman positions through this gate, and in the final instance, the sallying party returned but forgot to bar the gate behind them. The Turks noticed the unlocked gate, entered it and managed to capture the nearby tower, where they hoisted a Ottoman flag. This spread panic amongst the defenders and the panic amongst with the untimely death of the Genoese commander became the occasion that was needed for the Ottomans to take the initiative and capture the city.

The Emperor Constantine XI likely unknowingly also aided in the Ottoman capture of the city when he turned down the service of a Hungarian engineer called Orban/Urban who offered to make Constantine a revolutionary new weapon previously unknown to European warfare, namely the cannon. Constantine turned down Orban's offer because the cash-strapped treasury could not afford to either provide the resources Orban needed or to pay his high salary. The enterprising Orban then made the same offer to Mehmet II, who accepted, and it is agreed that it was solely the cannons of Orban that allowed the previously legendarily impenetrable Theodosian Walls to be breached.

I suggest reading some of the literature on the 1453 siege because it will be details like these that would help you on your project as the these terribly unlucky instances the Byzantines suffered during the siege are things you could potentially spin with.

Steven Runciman's narration-style prose of the Fall of Constantinople has that penchant of dramatics that you would likely appreciate for this project.

Runciman, Steven (1968). The Conquest of Constantinople: 1453. Cambridge: Cambridge University Press.