If MTG is trolling I'm interested in the psychological tricks to troll the trolls. But I usually hesitate to label someone a troll. I can't always tell who's trolling and who's not.
If someone really can't see the fallacies in their thinking that's a pity, but some people IMO use fallacies as a distraction, knowing some people will be determined to educate them about various fallacies and errors of fact. But if the errors and fallacies are deliberate, reason probably isn't going to help because the lies and fallacies are a well-calculated attempts to provoke a reaction.
MTG probably knows elected officials don't need to swear on a bible, but it won't help to point out she's wrong. Her remarks serve a dual purpose: to get the lie widely accepted by her fans, and to get a rise out of people who value facts and sound reasoning. So just say something peripheral like, "Omg, I sure needed to hear this. Does it have to be the KGV? I need to know because I'm buying Bibles for a courthouse."
(I wrote this because I think the remark about CrossFit is hilarious. Take what she says and run with it - but obliquely, in a direction she didn't plan on going).