The example that comes to mind is James Damore and he "doubled down" in the sense that he stood by his original statements (which made sense because his original memo was clearly and explicitly not endorsing the things he was accused of endorsing). It "worked out quite well" in that he _only_ lost his job and endured a lot of harassment, abuse, and slander (although who knows what kinds of psychological scars this treatment could have left him with), but probably didn't have a hard time finding another job and the mob did eventually (mostly) move on.
So either the Damore example satisfies your definition of "working quite well" but not mine (or probably most people) or this example illustrates that your prescription doesn't always work. Perhaps it is exceptional, in that most mobs aren't marshalled by major publications nor do they conspire with the CEO of one of the most prominent companies in the world.
He's in a tough spot given who he's up against. In an oligopolistic industry like ours it's risky to be hated by anyone.
But I do think it would work in the sense that he'd have a larger group of people cheering for him. And this is solely about clarity. If you're too nuanced in what you're broadcasting to the world people don't know how to react. Nobody wants to waste time trying to figure you out. Doubling down is exactly about taking out any nuance for a stronger message.
That's arguably one reason certain games companies have gotten away with horrible, rather anti consumer behaviour for years (like EA), because they realised early on that boycotts and internet complaints do not matter. For them, it's like 'who cares if the angry mob is calling for our heads over on Reddit, because Madden/FIFA/whatever will still sell millions of copies'
At the end of the day, most drama queens and controversy fanatics don't make companies money, and those that do will give up their boycott in time. So the answer is to do nothing and just watch everything fizzle out.
My advice to anyone would be don't make even the slightest effort to please people who hate you.