How do I approach a Protestant who "really" believes the conspiracy theories against the Church?
I’m not sure to which conspiracies theories you’re referring, but if you’re speaking to a Protestant, it probably involves the devil and that the Catholic Church is somehow a tool of Satan.
I would explain that the devil’s actual devotees consider the Church their real and main enemy precisely because Jesus founded the Catholic Church (see Matthew 16:18-19). That’s why they seek out the Eucharist—the sacramental “Real Thing”—from a Catholic church, instead of mere bread and mere wine or grape juice from a Protestant service. They realize that Protestants have merely a symbol while Catholics have the substantial body, blood, soul, and divinity of Jesus under the appearances of bread and wine.
In addition, Satanists perversely partake of Jesus in the Eucharist in their blasphemous “Black Masses”; they don’t reverently worship and partake of him in a genuine solemn Mass as Catholics do.
Further, I would argue that anti-Catholicism makes the strangest of bedfellows. Where else do you get such an eclectic coalition of people—including Hitler, Stalin, the ACLU, ISIS, Planned Parenthood, and the Ku Klux Klan—lining up together in agreement on a cause, despite their often great differences? And the secular media is no friend of the Catholic Church, either. If there were real conspiracies to expose, you can bet the wide-ranging and influential opponents of the Church would bring them to light, with the devil’s assistance no less.
In contrast, when a genuine scandal occurs within the Church, e.g., the recent clerical sexual abuse crisis, it's front-page news.
Finally, ask your Protestant friends to examine the Bible and history otherwise to reconsider whether the Catholic Church is the one, true Church Jesus founded. In that regard, I offer our booklet, Pillar of Fire, Pillar of Truth, which they can read online.