Mussolini wasn't really fond of Hitler in the beginning, so that is not a problem. The real trouble is to keep this dislike.
The main obstacles are two: Ethiopia and Spain.
The invasion of Abyssinia not only broke the so called Stresa front, but also caused the economic sancionts which in turn made Italy to shift her positions toward Germany.
The intervention into the spanish civil war deepened the division between Italy and the Allies (it was at this time that Mussolini ordered the navy to draw battle plans against the royal navy).
In order to keep Mussolini far from Hitler, you have somehow to circumvent these obstacles, keepingin mind that Mussolini wanted to demonstrate to the whole world how powerful Italy had become under fascism.
As for the South Tyrole, is not really an issue: if Mussolini and Hitler are enemies, I strongly doubt that the former would accept the anshluss. A war in 1937 would be quite a bloody and messy affair for both and I really don't know if Hitler would have risked his growing but still little and weak army so early. Something that changed quickly, but in 1937 Italy was still stronger than Germany. Besides I think that the allies, while avoiding direct involvment, could be tempted to give Italy some help to "teach the bloody huns a lesson or two".
Without Austria, Hitler would have troubles to annex Sudetenland, not to mention the whole Czech republic...
So with a Mussolini enemy of Hitler, you could either have an earlier war or no war at all.
in fact the first meeting Mussolini and Hitler in Venice 1934 is perfect for that
Hitler arrive with a J-52 From Berlin at Venice in wrinkled civilian clothes and soak with sweat
Mussolini and rest of Welcome Committee in full Military Gala Uniform and the Italian PRESS
Hitler was very embarrassing
after that He was taken direct to conference "room" a giant hall of a Venice palazo
here Hitler feel even more annoying in the extreme
note: Hitler had later let build by Speer a ever BIGGER hall as his office at Reichskanzlei in Berlin
here in Palazo Mussolini reamed about "Anschluss" Austria and the Death of Dollfuss
the dialog had to be Translated
Hitler look like cornered animal and ask about pause
during that pause Hitler demand that talks are made outside in Palazo garden
here he revived and take the advantage of situation wat let
to Mussolini estimation error "This man is not so bad after all"
so wat if Hitler din't get his way and stay cornered inside the hall
make fatal mistake to make agressive approach: "South Tyrole must back in the Reich or..."
and Mussolini squalls "OR WAT ?"
followed by Italian profanity wat are translated into German
followed by german profanity yell by Hitler wat are translated into italian
Mussolini lambasts Hitler (or is that last sentence a littel bit to ABS ?)