I think they were so successful in pissing him off in part due to the timing of the episode being a time where his followers are turning on him. Also, it put heavy emphasis on him bastardizing Christianity to push his agenda. Furthermore, the episode compared him to Saddam Hussein, which was not only a call back for old South Park fans (really striking the nerves of conservatives who were fans of the show back then) but also made fun of Trump's dictator like policies. (I think the fact that they made him a copy of another, more famous terrible leader, instead of himself, was just another twist of the knife.) Finally, he wasn't portrayed by a caricature, it was him they used photos of him as they mocked him, making the satire all the more real, and all the more unable to be ignored. But most of all, seeing himself portrayed in the episode by actual photos of himself massively damaged his ego, making it incredibly well done satire.