From what I recall Colonel De la Roque and his PSF were the only major far right party to gain any traction (particularly after the rise of the Popular Front and the dissolution of the parliamentary leagues even if they were excluded from the Government). I think one could accurately suggest that he was closer to the Gaullist model of a Presidential Republic. Also from what I understood there was a persistent fear especially among the Radicals but also from the Left more generally that women being influenced by the Church/more conservative meant that giving them the vote would be a boon to conservative and monarchist groups.@KingSweden24 Do you know colonel François De La Rocque ? IOTL he was a WWI veteran and french nationalist politician in the 30s, leading the Parti Social Français (French Social Party). He and his party had ideas like a stronger presidential regime to end the instability of the parliementary Third Republic, and despite being rather conservative, they wanted to give women voting rights and opposed antisemitism (De La Rocque make sure to exclude militants who didn't renounced antisemitism from the party).
I think he would just fit perfectly in Cinco de Mayo France !