A few points that could make them seem more Roman:
The Ottoman “Turks” didn’t even call themselves Turks (ethnically) until the 19th century, they actually used to call themselves “Rumii” (Roman in Ottoman Turkish, derived from Arabic).
Roman culture was not considered foreign or even looked down upon, since Ottomans themselves were ethnically called Romans.
Fatih Sultan Mehmed the conqueror embraced Roman culture, and was himself half Italian and half a bunch of other things.
Edit: fucked up the title, I meant continuation, not constitution. I’m on my phone, so I’m sorry about any spelling mistakes.
More can be written, but you might like to see the discussion "Why dont we view the Ottoman Empire as a continuation of the Roman Empire" by /u/mrleopards , /u/AStatesRightToWhat , and /u/Guckfuchs . Also, in "Was the Ottoman Empire the successor to the Eastern Roman (Byzantine) Empire?" , Searocksandtrees linked to older discussions featuring ConnivingToad and Ultach.
This is not to discourage discussion. More questions, data, and debate are welcome.
Edit: fucked up the title, I meant continuation, not constitution. I’m on my phone, so I’m sorry about any spelling mistakes.
Maybe I can help, I once did a study on roman imperial legacy during early ottoman period while I was residing in Istanbul. I focussed on the atmeydani, or horse square, a place in the center of the city where the hippodrome used to be. In a lot of ways the sultan continued the roman imperial tradition. Mehmet indeed saw himself as the successor of the roman emperor and as renovator of the roman empire. After the sack of constantinople in 1204 by crusaders the empire was in a state of decay. According to historian Rolf Strootman the ottoman conquest was a revival of a dying empire. Mehmet built his palace at roughly the same location as where the pre 1204 byzantine emperors resided, near what was left the the circus (commonly refered to as the hippodrome). The sultan saw the importance of this location, continuing to use it like roman emperors did, processions, war games, feasts. Mehmet also ordered his men, during the plundering of constantinople when he conquered it, to leave the hagia sophia untouched, showing his respect to one of the roman world wonders. Other mosques are even inspired by this building, like the famous Blue Mosque. However, the ottoman sultans were far more despotic rulers than roman emperors. For the emperors, a "healthy relation" with the people of constantinople was essential, which required visibility of the ruler (based on hellenistic ruler cult). Often the emperor would show himself and bless the people, for example before the start of the popular chariot races in the hippodrome. The roman empire derived from a free republican city state, where roman people were an important body in politics. The ottoman empire did not. They wanted to be Roman but it looks like they didn't fully grasp (or didn't want to) what 'roman' actually entailed. Most of the time the sultan refused to show himself clearly to the people, hidden away in his palace, or in a closed carriage. Also Mehmet referred to himself as padishah, meaning 'king of kings' in persian. Quite the opposite of augustus' first among equals philosophy. Anyway that was my take on things. I figure the subject is still highly debated.
Some literature: Brigitte Pitarakis and Ekrem Isin, Hippodrome/Atmeydani: A stage for Istanbul's history (Istanbul 2010). T. Kiousopoulou (ed), 1453: The fall of Constantinople and the change from the medieval to the modern age (Irakleio 2005).
Next ones are in Dutch, but I still name them anyway. Diederik Burgersdijk and Willemijn Waal, Constantinopel: Een mozaïek van de Byzantijnse metropool (Leiden 2011). Rolf Strootman, Mehmet Caesar: De val van Constantinopel en de doorwerking van de Grieks-Romeinse heersersideologie in het Osmaanse Rijk, Lampas (2013) 317-330.