As we’ve seen before, the Italian Fascists certainly weren’t afraid of oppressing civilians, and (as you can see in this paper) they usually had no respect for Jews either, so why the listlessness? Several reasons:

In occupied Yugoslavia and Greece, the Italian authorities had priorities other than the deportation or extermination of the Jews. They faced a very chaotic situation essentially because the Germans left them the burden of pacification. They were struggling against partisans in Yugoslavia and Andartes in Greece and their first priority was to restore order and annihilate the Resistance. Jewish refugees or communities, by contrast, did not represent a threat to the [Regio Esercito] on the spot.

Furthermore, to hand over the Jews to the Germans or Croatians would have harmed Italian prestige and alarmed the Četniks, who might have imagined that the same thing would happen to them. [Italian Fascism] desperately needed military collaboration with the Četniks.

Italian prestige, authority and reputation were frequently invoked by the junior partner of the Axis as the reasons why the Italian authorities wanted to pursue an autonomous policy with regard to the Jews, the refugees or any other issue concerning the occupied territories.

[…]

Along with these preliminary considerations, it is also important to emphasize at the outset that sources in Italian archives show no evidence at all of either a coordinated plan to protect the Jews or a conspiracy by the Italian Foreign Office and military leadership to disobey Mussolini’s orders.

On the contrary, both traditional élites and fascist establishment worked ‘toward the Duce’ until the beginning of 1943. Mussolini was always kept informed of policies and decisions relating to the Jews and very often intervened in decision‐making.

[…]

Finally, it is important to define the rather misleading term of ‘protection’. If one refers to the diplomatic protection that a sovereign state offers its citizens, the Italian government only provided such protection for Italian Jews in the territories annexed or occupied by the Third Reich; Italy provided no diplomatic protection for foreign Jews.

(Emphasis added.)