Perhaps France remains neutral in the Spanish War...
BUT : the Cagoule (which, OTL, had links with Italy) pushes harder in its terrorism campaign, or gots "lucky", actually managing to kill the whole government or even the whole Parliament with a well-placed bomb, or a very popular minister. While multipliating assassinations.
A new government (still Front Populaire) is constituted, and starts martial law. Moreover, all "suspects" are arrested, that include all known league members, and many entrepreneurs who were pro-Cagoule.
The official motive can be "terrorism" and/or "trahison" (since Cagoule is linked to Italy), plus association of criminals (this is actually punishable by law in France, I don't know elsewhere). Of course, all those people are to be tried, but in the meantime, dozens of thousands of people had just been arrested, and more had managed to flee (because they were warned by cop friends or things like that).
France enters a "witch hunt", because the Cagoulards and leagues have sympathizers everywhere, who help them to hide, and those people are also tracked. Including in the midst of police and administration.
Moreover, the ARAC (communist league) answers by its own agressions, threats of death...against real or supposed Cagoulards. The Party doesn't condone that but doesn't control his league either. And the ARAC gets a free pass from authorities (at most, some members spend a few weeks or days in prison before being released, and only if they did something REALLY bad), since the commies are in government, while the right-wing leagues are persecuted at the slightest offense.
Right-wing people are incensed, they find unfair that police turns a blind eye on communists but persecutes right-wing leagues, and put all the leagues plus the Cagoule in the same bag, while most of all never were terrorist.
So, France is polarized between the Front Populaire and the far-right. In the middle, republican moderate right tries to maintain its position without taking sides. They don't like Cagoulards (who are terrorists and traitors) but don't like what they see as Front Populaire growing authoritarianism. Especially as the French Communist is very strong in the Front, and they see (wrongly, Stalin never planned that) the roundups and witch hunt as prelude to a communist coup.
Perhaps, then, the Front Populaire government, fed up with far right (and its friends abroad, like Franco and Mussolini), decides to intervene directly in Spain.
They justify that with alleged links between Franco and Cagoule (I believe there were some OTL), instead of just saying "we take sides in a civil war", but nobody is fooled.