Well, to play the Devil's Advocate, I believe McCarthy himself was a Catholic, and a good portion of his support came from the Catholic community of the U.S., which was extremely anti-communist at the time.
It's important to understand that in McCarthy's day, the Church still exercised power over its flock, and there's still remnants of this American-Catholic identity today. My mother used to tell me about how priests would determine what movies you could or couldn't watch, music you could or couldn't listen to, so on. You were part of a community, and if McCarthy did start to go after Catholic Priests, even Marxists, he'd see his voter base gone and his community alienating him.
I think I've heard snippets of this conspiracy, but it wasn't explicitly described as "Communist."
So, there are two kinds of Catholic conspiracies: conspiracies
Catholics and conspiracies
Catholics. Chick Tracts are notorious for being some of the last vestiges of American Anti-Catholicism, in that he depicts the Church as this satanic institution that is simultaneously behind Communism, Nazism, and Islam. To some cold-war era protestants, Catholics were spreading communism so they could establish some Papal theocracy in the U.S.
From within the Catholic Church however, most conspiracies are that Satanic forces exist on earth (during the cold war, this was predominately Communists) and are attempting to undermine and destroy the Church. Extremely Right Wing Catholics are liable to believe that "bad" changes in the Church are the result of a conscious conspiracy by liberals/gays/communists to ruin its public image.
Central to both these conspiracies is the Society of Jesus - The Jesuits. Both within and without the Church, the Jesuits are probably the most controversial Catholic organization on earth. You either love them or you hate them.
In my experience, most of my Catholic school teachers had training by Jesuits, and we learned the story of Ignatius of Loyola growing up: a spanish noble who had his leg torn apart by a cannonball, spent his recovery reading the bible, and then actively sought to create a Church organization that ran off of military discipline and was willing to go anywhere to spread the word.
Now, Jesuits have a history of being Catholicism's most ardent missionaries. There were Jesuits in China translating Confucius, Jesuit missionaries won over a whole tribe of Natives by being tortured to death and happily accepting martyrdom.
Since Jesuits were trained in apologetics, they'd spend a great deal of time learning of the "Other Side's" viewpoints and then arguing against it. What this means is that they're both very smart, but also susceptible to liberalism given how often they experience diverse worldviews.
To conservative Catholics, the modern Jesuits are a "third column" seeking to push liberalism into the Church. To conservative protestants, the Jesuits are like the Vatican's equivalent of the CIA and are constantly working to undermine American values.
Anyways, there's an existing trend among Jesuits to be the more liberal clique within the Church... for better or worse. I believe a bishop with Jesuit training stirred some controversy recently when he depicted Jesus as a gay man. Similarly, I believe Fidel Castro was raised and educated by Jesuits.
So... yeah. This seems like a conspiracy pieced together from existing cultural conspiracies and also real trends within the Church.