I'm not sure keeping the Ottomans out of WWI altogether is necessary (although it is a fine idea) - I suspect the office of Caliph could survive losing a war if the Ottomans didn't get carved up and colonized. Is replacing Wilson enough, do you think, or were the British and French bound and determined to make a mess of things?
A spiritual-only leader of Dar al-Islam is not achievable through linear evolution, I don't think. The recent Pope's position is at least consonant with his Scriptures - there's nothing in there commanding him to be a temporal ruler and some things which could be interpreted as commands to eschew temporal power. The Quran and sharia (as I understand them; I am not a qadi), by contrast, consider and explicitly reject the separation of church and state.
A restored Sunni Caliphate seems very possible to me, but would seem to be primarily held back by a lack of candidates - the two most credible choices are the King of Saudi Arabia, who wouldn't touch the title with a 10-foot pole, and the heir to the Mogul throne, who leads a retiring and apolitical life in Delhi last I checked. Conservative Muslims in India and Pakistan do send him zakat (as a private citizen, his tax records are not public information, but he appears to recieve about 6 million USD per year, and give it away to Muslim charities without commenting on whether or not he is entitled to it). He's been invited to move to a more appropriate and respectful climate (Pakistan) and declined. Perhaps he has a more active and dynamic grandson? You could do a lot worse for a modern Caliph than someone who grew up in a democracy with freedom of religion, I think.
If you need him earlier, figure out a way to create an interregnum earlier and then "restore" the office.