Completed in 1938, as the Second World War raged on. When did it start? Was the Church ever in danger either directly or indirectly?
Given the modern political climate of say a company in Saudi Arabia having a tough time doing business in Israel, were things similarly back in the 30s and 40s where investments were restricted?
Edit: I have been informed that, in fact, 1938 was a year of peace and not war.
For the general purpose of your question, I think it is important to observe that in fact, in 1938, Europe was substantially at peace, and there would have been no obstacle for the Italian government to sponsor what was in substance a charitable initiative.
While I have not found yet details on the provenience of the funds for the project, this specific one was commissioned by the Custodia Terrae Sancte - The Custody of the Holy Land - after the discovery in 1936 by Franciscan Father and archeologist Camillo Bellarmino Bagatti of an ancient sanctuary, supposedly in the location of the famous "Beatitudes" speech.
The new church was built between 1937-38 by Roman architect Antonio Barluzzi. Barluzzi came from a family of functionaries of the Papal administration and soon after earning his degree, he moved to the "Holy Land" to take part and later supervise a series of projects for the various catholic religious orders active within the region.
That's exactly what the Custodia Terrae Sancte is: a branch of the Franciscan Order dedicated to take care of sanctuaries and assist pilgrims in the Holy Lands; and, as you can see, they are still active.
Under this perspective, it would be reasonable to assume that the Italian government may have chosen to sponsor this initiative, like it did with many similar ones, in the context of the persisting effort to establish - or re-establish after the hits taken with the Ethiopian War and the Spanish Civil War - the Italian international prestige. How much of the expenses were covered by the government and how much by the CTS I really can't say unfortunately. It does not appear to have been an especially remarkable event, at least given how none of the sources I have on hand seem to mention it.
More broadely speaking, the Middle Eastern region was certainly a crucial one for the interests of various nations; that went beyond the natural importance of Jerusalem as a religious center - where the Italian government had "sort of" an obligation to assist the interests of the Roman Church. The international agreements over the neighboring Suez Canal had been reviewed recently with the Anglo-Egyptian treaty of 1936, in what the Italians had taken as a direct reaction to their expansion in Ethiopia. But at the same time, Islamic independentist movements were looked upon favorably by the Fascist government as a possible anti-British instrument; there is somewhat of an extensive literature (and probably too extensive given the minimal results of such relations) on the relations between the Italian Fascists and the Grand Mufti of Jerusalem al-Husseini as well as the Iraqi leader al-Gaylani.
I do think nonetheless that searching this sort of connections here might be out of place. Unless I find something new somewhere, there is no reason to take this as anything more than government funding towards a charitable organization with the purpose of maintaining the Catholic presence in the region.