I am sympathetic to your wish for records about your grandfather, becasue my grandfather might have come from Syrna as well, but he settled in Alexandria, Cairo. Your Antonio might well have arrived at the end of WWI (you say 1917), but trouble was brewing in Smyran to n all out invasion of the Greeks on Turkey in 1919, which resulted in atrocities. The Turks got their revenge in 1922 by burning all things Greek and Armenian in Smyrna (a huge percentage of the city) where a park now lies. You will not find vital records of Greeks there.Most Greeks who lived there were either killed or forced to flee Smyrna, or later traded for Turks with Greece.
Church records might exist in Istanbul, where the Patriarch of the Orthodox Church is. That would be interesting to investigate, for myself as well. There also is a small suberb in the north of Athens that is called Neo Smyrna. On the main square is a library.In the library is a set of books called in the Chronicals of MicroAsia, all in Greek. It is an attempt to assemble recollections and facts about Smyrna before the disaster of 1922.I found my grandafathers surname and an actual map of the street and address where that family lived, there were a couple. I could find nothing that would connect my grandfather directly to them.There might be more. A person I hired actually did the research on this for me. I have no how thorough she was able to be. More digging there might have produced more. If you need more specifics about this place, please let me know.