Have the scandals of Catholic Priests molesting children happen earlier and make the public learn about it in a Watergate like event in the 60s or 70s.
I think that's about the only thing that would work, and "Watergate event" would probably have to be something as high-level as Watergate itself.
Which might be difficult, because I don't think federal jurisdiction over child-abuse cases was a thing until the last thirty years or so(open to correction on that). But maybe have some big-city Irish mayor somewhere squelch an investigation into an abusive priest. And then AFTER he enters the national arena(maybe as a senator or even veep candidate), his earlier cover-up is in danger of being revealed by the press and/or law enforcement, so party bigshots use illegal methods to stop it from coming out, and then THAT cover-up gets exposed, and you've got your scandal.
But even then, I don't think it would lead to the long-term survival of Know-Nothing anti-Catholicism, maybe just keep it on life support for a few more years. The backlash against the RCC would probably just get subsumed into general 1960s anti-authoritarianism, which most baby boomers are going to outgrow once they turn 30 anyway. And the true heirs to the Know-Nothings are gonna be in the Religious Right, who by the late 1970s will still be inclined to ally with Catholics on social-issues like abortion and gay rights.