What do I do? Do I simply begin looking for other positions?

If you mean external other positions, the answer is yes.

Looking for internal transfer is not going to be effective. Once on a PIP, your odds of moving are effectively zero. No one will say that transfer's impossible (legal/HR reasons) but it's basically over. It sounds like you were a good worker but broke the #1 Rule of Corporate Employment: make the boss happy. Don't worry; this is a common mistake.

Do I report his increasingly erratic behavior, and waste of firm resources?

I wouldn't. It's not worth your time, and it won't get you your job back. You have 30 days to live, and it takes months to beat a bad boss. Not only that, but it takes reports from about 3-5 subordinates and most people aren't willing to risk their jobs. Corporations value consistency over correctness and his being a manager means that they trust him over you. You won't win. You'll hurt him a little bit, but that shouldn't be your goal.

I know someone who actually beat his boss. He had a bad boss, he got a bad performance review for something that wasn't his fault, and he managed to get five or six people to hit his boss with the same story in "360 reviews" and take him down. It took months and, while his performance reviews were amended, he was still screwed. Even though he had taken out one of the worst managers in the company, managers (like police) tend to protect their own, at least when it comes to the outside world, and no one wanted to promote him or take him on. He ended up getting laid off a year later.

Do I quit before the period expires?

Only if you find another job. You might get a severance if they fire you. If you quit while you don't have a job, you get nothing and the result (you're unemployed) is the same as if you'd been fired.

You are going to get fired at the end of the 30 days, but getting fired isn't (except in a few overly chatty/cliquish industries) a big deal at all.

I do not wish to move, and if fired I will be effectively blackballed from most firms in this city.

Why? Most people, by age 40, have been fired at least once. It sucks, but it's not the Scarlet F that you might think. If you pass the siege test (6 months of savings) then it may be expensive, but you'll survive it.

As for reputation issues: I think the firm will be good to you unless it's run by idiots. You have knowledge of illegal activity. You absolutely cannot leverage that in asking for a cash severance (it's extortion) but, if they don't agree to a contractually-guaranteed positive reference, it might be time to bring up the fact of the need to reach mutual agreement on a story that is beneficial to both parties. You'll probably get an arrangement in which you can represent yourself as employed during your job search. You may get a contractual written reference and an agreement by your manager not to contradict it.

They would rather let you go in a way that doesn't blackball you than have you blow the whistle. Trust me.

Use your colleagues as references and have your manager's reference checked by a third party, even if you have a contract. Unless his reference is glowing, don't use him. Say that your manager went on a mission to Uganda and that's why you can't use him, or that you were managerless due to bureaucratic error but colleagues X, Y and Z can speak for your ability. Or, take the risk and tell the truth: you refused to do something illegal and were fired for it. (Say as little as possible; bad-mouthing an ex-employer is generally a no-no, and it's usually seen as better for both sides to tell a mutually face-saving lie than to expose an ex-employer's bad behavior.)

You should reach out to me (michael.o.church at Google's email service) if you still have questions.

Getting fired is bad-- three months of joblessness is expensive, especially if they fight your unemployment claim-- but it's not catastrophic and the likelihood that any of this follows you is low.

Also, it's actually pretty hard to blackball someone. I've had people try to blackball me and I ended up winning. In some cases, I've made the attempts to do so very expensive for the people involved.