My personal preference is to always respond first in good faith, even if the shit clearly stinks. Sometimes they just worded things poorly or misunderstood something. However, if their stench becomes apparent, it's then much easier to humiliate them and dip by the second response.

I feel like, if anyone calls out a fallacy and acts like it's an "I won" card, you should just pull out the Fallacy Fallacy and uno reverse that shit. Then fuck their dad as a victory lap.

To go a step further, arguments are healthier if they're pictured as a way to field test your beliefs to see if they hold up to scrutiny.

If you go into an argument trying to get the other person to change their mind, you'll often be met with failure even if your points were valid simply because people hate changing their mind, and you don't want to be tempted to use bad-faith arguments of your own just to secure that "win."

Instead, just give your argument; if the other person has a good point, see if yours can hold up to it, and change your outlook if you find that it can't. And if it feels like the other person is just saying whatever they think will "win," leave, because their argument wouldn't make a good field test anyway.

i call out specific logical fallacies when my dad is going off on some bullshit and i want him to set aside his gen x jaded cynicism and actually listen to what I have to say. usually works pretty well but thats because he has critical thinking skills, which one would expect to be a prerequisite for a debate or even a stimulating dicussion but oh well.

Calling out fallacies isn't done for the benefit of the muppet. It's done for the benefit of onlookers who might otherwise fall for the muppet's bullshit.

the sort of person who believes with the brain but not the liver

Meaning that they’re less likely to spit bile in their defense of religion, or is this a reference to some religious belief?