My take is a sort of dysfunctional republic of oligarchs is more likely. Assorted factions of the wealthy squabbling over this residual impoverished Turkish state. Some sort of facist party or doctrine may emerge, but I cant see this state as of any practical attraction to Germany or Italy in any major way.
I'm also looking at the potential for Greece in this. First off there is the cost of asorbing the Ionian refugees in the 1920s waived away. Second there is the economic benefit of owning Iornia. The net gain from these two is large in the context of the Greek economy in the 1920s.
A third item affecting Greece is the status of Constantinople through to 1940 & beyond. It is not impossible Greece can gain control of that territory later in the 1920s or thirties.
Then there is the situation of the other territories sectioned off in the Severes treaty. Who retains control of those through the 1930s? In 1941 who controls the Chromium mines, & other critical mineral deposits in modern Turkey?
seems the most straightforward (realistic) change would be Greece could hold E. Thrace/European Turkey and (British) International Zone over Constantinople? (with the Turks glaring across the Straits)
with all parties exhausted none of the other aspects survive? that more closely mirrors historical situation?
I can see that as a likely outcome. I have now doubt the Greeks would make plans to secure the Constantinople region. But, the ability to execute any plans is a large variable. Very situational.
my reading on the subject is limited. it seems a few gambles by Greece decided the outcome? after they were evicted from Asia Minor they would need propping by Allies to hold Eastern Thrace.
maybe the Allies just never help them land there to begin with? not aware if Cyprus could ever be a bargaining chip for UK? or the Italian-held Dodecanese, at one point their return was promised to Greece. just seems easier for Greece to acquire the islands and contiguous land mass (E. Thrace)
seems far from certain Turkey would become a Fascist ally? there might even be a side in Greece supporting revanchist Fascist rule? if Smyrna ends up as Free City type arrangement or more likely Greeks forced out?