Seems to me like it's a matter of gauging the other person's willingness to change their mind (an admittedly difficult task in itself). If there is no argument or evidence that you could present which would be good enough to do so, the entire conversation seems to be pointless and it's probably best to just walk away. If they are in fact open to being proven wrong, doing so politely may still not work (i.e. they may be able to counter any point you make), but the odds are better than if one does so rudely. All other things being equal, we're more likely to be persuaded by someone we like (or at least are neutral toward) than someone we dislike.

> Now, how long do you think you can last being polite?

Difficult, I admit. I don't have tons of patience with the clearly-wrong-but-backed-by-spurious-arguments-and-willing-to-argue-forever types.

> More to the point, WHY?

At least in the case of the doctor, because if the person is your patient, it's part of your job. And even if they aren't, it's still a health concern. Reach them if you can.

Now, there's a place for a verbal smackdown if you're at a party, say, and this person starts telling the room their anti-vax baloney. It's important that the room not believe this person. But still, be as polite as you can.

Right. In these contexts my reply is usually “fuck off”. Similarly with people arguing in bad faith. Life is too short to waste time with lunatics, but at the same time there is no point attacking them.

I have seen few complete nutcases in a scientific context, though. But sometimes decent people have strange opinions about, say, vaccines, in which case I tend to change subject or walk away.