The OP states that Italian neutrality is of the type Franco adopted IOTL.
That's a rotten analogy though, the cases were very very different.
If Mussolini is "pro-German," there's little reason--given the poor foresight of the Axis leaders (except maybe Franco, if we count him as Axis at all) for the long term prospects of war with the liberal powers and Soviet Union--for him not to go all in with Hitler and hope to ride his coattails, plus of course Mussolini had a far more sanguine view of Italian capabilities than seems justified in retrospect.
A better scenario would be Mussolini is really pretty hostile both to Hitler and Allies, and decides Hitler is the lesser evil but will not risk getting entangled with him either. Entirely plausible IMHO! Mussolini did resent Hitler's snatching Austria from his hegemony (via his patronage of Dolfuss) into the Reich. Assuming that is going to happen, all it takes is less Hitler kissing up to Mussolini. That's the way it worked initially; Hitler admired Mussolini for inventing Fascism and succeeding in the 1920s when he failed then. Nation to nation, the alliance was poison for Italy but it apparently tasted good going down. Dictator to dictator, Hitler could have left Mussolini out to dry but did not.
Just sour Hitler on Mussolini, let admiration turn to envy before Hitler has a chance to prove himself, and hostility on that border is offset by the hostility of Britain and France to Mussolini's East African ambitions.
Being caught between the two sides, Mussolini decides to be coldly neutral to both and pursue as much as he can get away with without annoying either side into intervening in southeast Europe.
No, I don't think Hitler focusing on taking Russia from north to south instead of advancing on the broader front will enable him to deliver a knockout punch to the Soviets either. Even assuming Hitler can muster the same resources, he still will want to enlist Romania as ally if he can, and the Romanians might risk annoying the Italians to do it anyway. If perforce, due to a neutral Romania playing off all three sides (German, Italian, Soviet) successfully, Hitler must concentrate forces on the Polish and Baltic fronts exclusively, I still think any multiplication of success in the north is offset by not taking the south, and many of the German targets they most desired were in the south of the USSR anyway.
Basically either Mussolini can persuade the Western Allies to rupture with Stalin before Hitler is completely sorted out and becomes a big player in a right-tilted NATO like force with him having a freer hand in southeast Europe (those parts the Soviets don't take, which might leave him practically nothing there) and retains all African holdings (until the Africans throw them out eventually anyway) or if not, the Soviets take most of SE Europe with Allied blessing re spheres of influence and Italy is crowded out of Europe outside Italy itself. Postwar Fascist Italy becomes an anti-Soviet ally.