Just a point of clarification: How precisely would Austria-Hungary invade France through Italy? Don't you think the Italians would have something to say about this?

The nature of Italy's accords that existed between Germany and Austria-Hungary in the lead up to the first world war is often misunderstood: although diplomatic discussions with the German government had been in the works over the course of the second half of the 19th century in order to lay the groundwork for a German-Italian alliance that would guard against French expansion into Tunisia and Libya, Germany’s close relationship with Austria stalled these negotiation until France moved to occupy Tunisia in 1881, after which the panicked Italian government swallowed its pride and immediately pursued a rapid normalization of relations with Austria. By 1882, Italy had turned the Austro-German “Double Alliance” into a “Triple Alliance.”

This outcome was far from a foregone conclusion, nor was it an entirely predictable turn of events: the Austrian Empire had been the primary antagonist in the past century's conflicts that characterized the emergence of the new Italian Kingdom. Indeed, in spite of the "Triple Alliance," the Italian political community also worked towards finding a diplomatic solution with the French, and by 1900 a treaty outlining the respective spheres of influence in North Africa had been signed. Thus when news of Archduke Ferdinand’s assassination reached Rome, the reaction was one of guarded indifference, while the Austrian ultimatum to Serbia in response was viewed with unabashed surprise. The Italian government was not about to jeopardize its delicate agreement with France in order to help a country it had only reluctantly normalized relations with.

Further, the Triple Alliance between Italy, Germany, and Austria-Hungary, beyond being a fundamentally defensive treaty, also required that all parties keep each other informed of their diplomatic and military initiatives. However, the Austrian administration knew full well that Italy would never be enthusiastic about an Austrian-led war in the Balkans; as such never moved to make the case arguing for the “Defensive” activation of the treaty, and didn't even inform the government in Rome about its plan to deal with Serbia. As such, when a state of general war broke out in Europe, no one batted an eye when Italy declared neutrality.

Indeed, the Triple Alliance had specified that all Austro-Hungarian expansion in the Balkans would necessarily need to be matched with "concessions" to Italy, which were generally understood to mean the cession of the majority-Italian regions of Trentino, Istria, and/or Dalmatia. For Italy to concede anything to Vienna, or as specified in the question allow the Austro-Hungarian army to march across the Po valley to meet France (and likely turn its own terretory into a battleground) the Italian government first would want to know what, precisely, the Austrian government intended to cede. If it there was no commitment to these concessions, the Triple Alliance treaty might as well get torn up.

So rather than jump to anyone's side, over the course of 1914 Prime Minister Salandra oversaw a cautious debate over Italy’s role in the war. General Cadorna, a Senator and the head of the Army, warned that in the case of a Germano-Austrian victory (the most probable outcome without Italian intervention, argued by Cadorna and the Foreign Minister Antonino Paternò Castello) Italy would certainly face some sort of a reprisal. However, both Socialist and Catholic members of Parliament were against intervention; although in the minority, they were able to force negotiations to be opened with Vienna to ask for precise cessions in the event of Austrian expansion even if Italy didn't participate in the conflict (after all, that's what the Triple Alliance agreement said). The Austrians agreed in principle, but insisted that Italy first join the conflict before any precise concessions were agreed upon, thus bringing negotiations to a halt.

An interesting and telling development is that upon the death of Minister Antonino Paternò Castello in the fall of 1914, the Foreign Ministry was entrusted to Sidney Sonnino, a peculiar man and a former Prime Minister. Sonnino, whose mother was British, was an anti-Catholic conservative who was in favor of intervention on the side of Berlin and Vienna (going to far as drawing up war plans to this effect with Cadorna). But in spite of Sonnino's position, the mood in Parliament held that it was only a matter of time until Austria and Germany turned on Italy! Italian neutrality, the general feeling in parliament held, could only be guaranteed if there was some sort of binding neutrality agreement signed with Vienna.

But in the end, the promises that the Allied Powers could make were much more appealing than any vague promises of "concessions" that Vienna was willing to offer. In late 1914, the Russian Foreign Minister Sazonov offered Italy Trieste, Trent, and the Albanian port of Valona as prize for a successful Italian intervention against Austria. Although the Italian reaction was not warm, seeing as the Russians had offered portions of the Dalmatian coast to Serbia, Italian satirists began poking fun at the way the whole of Europe seemed to be courting Italy.

What really warmed up the Italian political and popular audience to war was a series of “Baby Steps,” all in favor of the allies. First, the Austro-German alliance brought the Ottoman Empire into the fold, peeving the Italian political class that had been planning the Ottoman downfall as a fanciful side-project. In response, the Italian-backed anti-Ottoman revolt in Albania turned into an outright occupation of Valona in order to prop up William of Weld as Prince of Albania. Salandra’s end-of-year budget allowed for an increased defense expenditure: it would seem that as the situation in Europe degenerated, Italy had no choice but to prep for war (the Italian press received the news rather ironically, joking that Italy had been seduced by another woman, abandoning Trieste for Valona).

Although Sonnino again invoked the Italian right to “Compensation” as per the triple alliance treaty, the Austrian government now asked to delay talks until the front stabilized and the occupation of Serbia was completed. The German government seemed worried, and appointed former Chancellor von Bülow to the post of Ambassador to Italy, with the dossier of keeping Italy neutral. But although von Bülow seemed interested in bringing the Austrians and Italians to the negotiating table, Austria was unwilling to commit to compensations until the war’s outcome could be discerned. This finally made Sonnino turn his position and advise Calandra to open negotiations with the Triple Éntente.

So you see, although Italy had initially started the war closer to the Austro-German camp, the underlying tensions between Italy and Austria (both competing in the Southern Mediterranean) made any sort of agreement impossible.