The challenge: with a POD after the Congress of Berlin, have an independent Kurdistan come into existence, that looks at least roughly like this.
Probably doable in the aftermath of World War I, created in the breakup of the Ottoman Empire, though the trick is getting the Kurds noticed by the Great Powers. It likely wouldn't be independent at first (League of Nations mandate?) but could gain independence later on.
(wonder who this would be? Did Kurdistan have anything resembling royalty?).
In 1880, a few years after the Congress of Berlin, Sheikh Obeidullah lead a revolt by ethnic Kurds. That revolt failed because he ended up fighting both the Ottomans and the Persians. Now let's say the revolt ends up differently. Perhaps the Persians or Ottomans screw up, or Obeidullah makes better decisions, or just gets lucky during the battles. End result is either an independent state, or perhaps an autonomous region within the Ottoman Empire - perhaps by combining the Kurdish areas of the Persian Empire within the Ottoman.
In either case, it would help protect an incipient Kurdish nationalism as ethnic tensions rise in the twenty years afterwards as a result of the Young Turks attempting to establish a centralized state based on ethnic Turkish identity.
This would give you a core area of nationalist aspiration and a leader who could negotiate with the British and French at or after the end of World War I. Then in the period of November 1918 and June 1919 when the time was most favorable for the creation of a Kurdish state, there would be a leadership clique who could achieve it. IOTL, effective national leadership had not yet occurred. ITTL, with an existing Kurdish lead satapry as a result of the 1880 revolt, there would be such.
The Kurdish leader is able to impose effective control over much of the Kurdish area. The British in particular are receptive and recognize Kurdish independence as do the French. Confronted with the Allies, Greeks, Armenians, and Kurds, the Turkish leaders accept the Kurdish claims. Britain does not attempt to place southern Kurdistan (Mosul/Kirkuk area) into Iraq, and recognizes it as part of Kurdistan. Since the Kurdish areas of Persia already became part of Ottoman Empire back in 1880, even this region is part of the new state.