Tricky but interesting
I've been immersing myself in this period researching a French Civil War scenario, so this drew my eye. And I can conceive of a twist here and there that would have led to such a war, which likely would have been brief and vaguely comedic. Let's face it, we've seen nations go to war using armed Toyotas. If two groups of rival political elites get mad enough, there is no limit to how willing they are to go tell the lower classes to die on behalf of their temper.
As an aside, one of the frustrating aspects of the site to me as a reader is that when someone says, "What if --" s/he is buried under reasons why it could never happen, citing all the reasons it didn't. Sometimes an idea is absurd, but if there is some plausibility, I enjoy seeing what would happen from there. The fun really is in the *result* and *consequences*, not the POD.
So go for it.
I'd suggest two big obstacles to address. First, neither country had a great deal confidence in its military readiness. France had some wonderful toys to play with but no idea what to do with them. More critically, Gamelin had no real interest in figuring it out. French military history in the 1930s consists largely of commissioning staff studies that proposed reforms that Gamelin would automatically reject as involving just too much work. (The legend, earned or not, of Charles de Gaulle began here as his proposals on the use of France's top-of-the-line tanks were rejected by Gamelin who really didn't care to stop brooding and sighing heavily long enough to seriously study them.)
As for Mussolini, he spent the first part of the decade plotting to retain his World's Top Right Wing Dictator laurels by out-maneuvering Hitler in reference to Austria, among other prizes. His methods were entirely grounded in ideological appeals (he was a rare bird, an "international" fascist) made via his chief diplomat and son-in-law. Ciano cut a dashing figure in a tux, and by the time he became Foreign Minister he had mastered a haughty sneer almost Parisian in its precise and devastating execution. Mussolini seems to have assumed these things would carry the day.
Hitler was more of a cut-to-the-chase kind of despot. Spotting Benito's game -- rouse up the Austrian fascists to take Vienna while cutting the Austrian Nazis out of the game -- he may, perhaps, have paused ever so briefly to contemplate diplomatic maneuvers of his own. In any event, if he did, he quickly waved that off in favor of marching into town himself, at which point Mussolini's "Austro-fascist" group vanished.
When his rivalry with Hitler began to degrade into humiliating sycophancy, he was forced to advise the Führer his own armed forces weren't prepared to take up co-belligerency in any competent manner. That was in August, 1939. And as was subsequently, repeatedly shown, the Italian Army was badly equipped and even more badly led. (Rommel, who fought as closely in coordination with the Italians as any German general, commented that the found the Italian soldier admirable, his equipment deployable and his leaders laughable.)
Second, the French were just as reticent as the Italians to engage in war, not because of a poorly equipped army, or bad field leadership, but because the see-saw of French politics in the 30s meant that French foreign policy consisted (insofar as it consisted of anything) of "wow, sure hope the Germans don't invade us." Well, to be fair, there was one other consistency in French foreign policy of the 30s, which was promising the Poles they would go to war to defend them from Germany while whispering to the Germans not to take that too seriously. Which the Germans didn't. Hence, Hitler's fury with Ribbentropp when -- as much to their own surprise as anyone else's -- the French declared war after the invasion of Poland. Funny world, this.
Sure, there was a then tiny group of French nationalist-socialist thinkers and advocates who admired Germany and insisted the real threat was Britain, but their influence in the 30s was de minimis. The sudden embrace of Nazism by French fascists post-surrender was -- just that. Post-surrender.
The French people were also -- understandably -- exhausted. They'd dealt with German invasions twice in less than a century's time and been badly bloodied. They had no real assurance England, let alone America, would help them out, and were more than a little bitter that it was their pan that always got burned black, war after war, while Brits and Yanks showed up, fresh and re-supplied, kicked the Germans across the Rhine, and then demanded the French grovel out some thanks for saving their bacon.
To understand just how lethargic Gamelin's French Army was throughout the 30s, remember that when invaded by clouds of German planes and hoards of Panzers, Gamelin's advice to the Prime Minister was to sit back and wait and see how things developed -- best not act rashly. Or at all. And in the 1946 hearings, Parliament finally answered the question of why the French Air Force, which was not at a major disadvantage on paper, failed to show up and contest the Luftwaffe's sorties or bomb German troops. Two things: (1) le Armée de l'Air's commanders refused to order their planes into the air, fearing they might be shot down -- better to save them for a rainy day; and, (2) the newer fighters, e.g., the D.520, had been delivered to their respective fields well before, but no one could be bothered to attach the propellors, and when these were finally located the Hun was at the gates, airfield by airfield. (The D.520's turned over intact became part of Axis air forces across Europe.)
Summarily, it's hard to imagine either country committing to invade the other.
But it could have happened, precisely because war is almost always the result of irrational decisions made on imperfect intelligence.
I agree with one of the earlier commentators that France would have to be assured Germany would not attack. The actual balance of forces doesn't matter. After the war, French generals and intelligence officials admitted they conflated the figures in order to convince the government not to get too adventurous. And, as so often happens in such a situation, even *they* started to believe them.
A deal might have been cut, though, in the middle of the decade, because Hitler was aware he needed a few more years to build his forces, and he was aware of Mussolini's intrigues against him. Remember, this is also during a period that the French always assumed Hitler would keep a solemn oath derived from diplomatic conference.
