With the Germans and the Bulgarians being as stretched as they are, I don't think they want to make matters worse for themselves by turning another country against them. Then again, they did blunder into war with America through stupidity and wishful thinking OTL, so I guess we can't exactly dismiss it as impossible.Pushing stuff across Anatolia to the Caucuses is easier than OTL taking it around the Turks and you are right that the Germans/Bulgarians might well push towards Constantinople and even take it though that might paradoxically make the Allies life easier by turning the Turks from un-cooperative defeated foes to sort of Allies.