The problem is that defending against a copyright troll in the court is an expensive headache, and the copyright troll has a whole army of lawyers to prove for sure that the Moon is made of green cheese. As such, even if the target knows that it's a bogus claim, they still comply with the troll to avoid the court.

Sending a takedown notice under DMCA that’s knowingly false is perjury, which would presumably come up at the court hearing.

In theory, yes. In practice, good luck proving that the copyright troll knew it and acted maliciously.