um, I don't think that the Pope and the Church is going to be pleased over the Italian King taking that title, especially since it was the Italians who destroyed the Papal states just 40ish years ago. Im not too positive but I think the King was actually excommunicated during this time, as Im fairly sure Catholics were encouraged to not participate in the Italian government at the time, due to the whole "prisoner in the Vatican thing."
The Savoy royal family has some legitimate claim to the title of king of jerusalem so is not really out of thin air; regarding State and Church relationships...well is complicated, yes officially the Catholics church at the time had that stance regarding italians politics but from the start of the XXth century both side tried a akward rapproachment (the last Giolitti goverment before the war was done with the support of the catholics) so by the time of the war there were many 'private' and 'unofficially' catholics political association. Giolitti goverment coalition is at the moment an akward, strange and in many sense fragile grand alliance of liberal, moderate socialist and moderate catholics keept together by the policy of neutrality; and if the Ottoman gave to Italy the 'work' of protector of the holy site/religious order so to 'spite' the Entente, the Italian goverment accepted as a conciliatory move towards the papacy and the catholics...basically the ITTL Latheran pact will be probably signed a little earlier.
