Actually.. I don't think a Emigrés or a foreigner, being installed to head a government will have popular support.whether it is Republican or Monarchist..
While this is true (it was true for Sun Yat-sen and Syngman Rhee) for individual leaders, I think that the Republican government as a whole may be able to accustom itself to its surroundings and outlast its founders.
Of course, it's not that Germany planned to wait in Anatolia forever - one day or another, they hoped to either return to finish the job or give the occuped territory should any successor regime be comfortable enough for them. What the Germans couldn't have expected, however, was that the front line would soon become a hard border between two rival regimes.
I'm spitballing here, but if the border between German-occupied Anatolia and the Yenilemists becomes a hard border then:
1. The Germans create some democratic "West Turkey" in their occupied lands (maybe they even make it a German-style constitutional monarchy ruled from Constantinople under a relative of the Egyptian line) and the Yenilemist "East Turkey" coexists uneasily with it.
2. The Germans give all of their territories in Anatolia to Greece, which must defend and colonize them despite the protests of the Yenilemists, the sole remaining Turkish government.
3. The Yenilemists are defeated by the Unitarians, the Germans stay in Anatolia, and the border becomes a German-Indian one.
Overall, I'm REALLY not sure if I like Germania's idealism that much. The declaration of a new war immediately after the last one doesn't feel like a good idea. Plus, Germania's insistence on immediate and meaningful transitions to democratic government is nice but some of the most successful democratic transitions of OTL (Taiwan, SK) came after years of tolerated dictatorship, years of relative peace (or at least, the absence of civil war) and prosperity in which local economies could grow interlinked with the world, native educated classes with a sense of civic nationalism and duty could develop, and movements for democracy could be created. If a democratic German-backed country starts turning illiberal or authoritarian, what then? Will they let the locals fight for themselves, or will they immediately intervene with election observers, sanctions, or worse?
Once it has ended the war in Eastern Europe and started designing international law, Germania (and Europe at large, and the world) had better have good answers to very tough questions.