Not sure how that follows.
A decision to allow Hitler to annex Austria does not commit the Duce to war when it comes. And, in fact, it certainly did not when Hitler kicked off WW2 by invading Poland in September, 1939 (nor did the Pact of Steel!!). Mussolini certainly did not seem to see it this way.
Mussolini might even reason it out this way: "In the long term, Italy does not have the strength to keep the Germans out of Austria. If I give Hitler a formal permission now, he will be in my debt; it will buy us some forbearance, while I pick my time to intervene." Of course, Mussolini had a more immediate motivation in mind for giving Hitler a green light: he wanted German support for his consolidation of Ethiopia. Which he got.
And the truth is, Hitler really did give the Duce that forbearance, in spades. No matter how many other treaties or agreements he broke, no matter how many promises he betrayed, Hitler really did stick by Mussolini to the very end. Recall the transcript of the meeting where Mussolini, communicating through Prince Philip of Hessen, the German Ambassador at Rome, gave Hitler permission to invade Austria:
"F (Fuehrer): Then, please, tell Mussolini, I will never forget him for this.
"F: Never, never, never, whatever happens. I am still ready to make a quite different agreement with him.
"H(Prince Hessen): Yes, I told him that, too.
"F: As soon as the Austrian affair has been settled, I shall be ready to go with him through thick and thin, nothing matters.
"H: Yes, my Fuehrer.
"F: Listen, I shall make any agreement.I am no longer in fear of the terrible position which would have existed militarily in case we had gotten into a conflict. You may tell him that I do thank him ever so much, never, never shall I forget that.
"H: Yes, my Fuehrer.
"F: I will never forget it, whatever will happen. If he should ever need any help or be in any danger, he can be convinced that I shall stick to him whatever might happen, even if the whole world were against him.
After the invasion, when in Linz, Austria, Hitler communicated his gratitude to Mussolini once more, this time by telegraph: "Mussolini, I will never forget you for this."
It is hard to see how things would be much different in a timeline where Mussolini maintains a very actively friendly non-belligerency, acting as a conduit for all sorts of scarce resources from abroad. Not least because OKH will be more than relieved to not have to be dragged into any side escapades on behalf of the Italians, whose military effectiveness they (and indeed Hitler) did not rate highly.