Some Monarchists, even coming from the ranks of the Action Française, did actually support the Resistance. Because even though they were Monarchists, a good deal of them were also highly nationalists and didn't like the idea of a german occupation and some even less of a german cooperation.
It is generally considered that during the Occupation, the Action Française actually divided into three groups: the Orthodox Maurassians, the Collaborationnists and the Resistants.
The Orthodox Maurrassians were those who actually followed Maurras in his choice. Maurras did support Vichy but that had more to do because of his respect for Marshall Petain and because Vichy's policies were generally pretty close to his own vision for France. He also has a great disdain for the Resistance because of his Anglophobia and his hartred of Communism, as well as his own desire to avoid a civil war. However, Maurras was an even stronger Germanophobe... To the point the Nazis actually put him under their list of people to keep under watch. And he also clashed several times with proeminent Collaborationnist figures like Laval or Marcel Déat. He even ended his friendship with the writer Robert Brasillach when the latter decided to openly support the Collaboration. It should also be noted that Maurras was actually no fan of the Nazis: before WWII, he had actually wrote down a number of articles were he considered that Hitler's ideology was one of the most dangerous in the world. And even though Maurras was antisemite, he apparently never went as far as to advocate murder, so the Shoah kinda horrified him as contradictory as it seems.
To sum up, you could actually classify Maurras and his most Orthodox followers as pro-Vichy but anti-German. This kinda puts them in a "neutral" area during the conflict.
The second group were those who openly collaborated with the Germans. Among them are several of the people linked to the highly pro-Collaborationnist newspaper Je Suis Partout: the writer Robert Brasillach, the journalist Lucien Rebatet and the editor Charles Lesca were former members of the Action Française. They all happened to have broke ties with Maurras afterwards though because of the Pro-German views (and Rebatet actually went on to write several articles decrying Maurras' views). Another figure was that of Louis Darquier, the man who ended up becoming head of the Comissariat aux questions juives: he started as a member of the Action Française. And finally, there is Joseph Darnand, the infamous leader of the Milice, once a member of the Action Française. A lot of these people seem to have left the AF before the War however, but that doesn't mean you didn't have a portion of the AF that actually followed their example and became openly Pro-Nazi during the Occupation.
And finally, the final group is that of those who joined the Resistance. You have for example of Count Honoré d'Estienne d'Orves, descendant from a noble family with an undeniable Monarchist background, who joined De Gaulle in 1940: he would eventuall be arrested and executed in 1941, after he had been sent back to establish a Réseau in France. Other names I could mention are those of Michel de Camaret, Henri d'Astier de La Vigerie, Gilbert Renault (better known as the Colonel Rémy), Pierre de Bénouville, Daniel Cordier or Jacques Renouvin. All of thsese people were from Monarchist background, some having even joined the King's Camelot before the war. And all of them joined the Resistance, with a good deal of them having joined as early as 1940.
So basically, having French Monarchist join the Resistance wouldn't change things much considering they actually did...