The solution is to go nuclear and enshrine adversarial balance in law.

If a company, or an agent acting in their name, makes a fraudulent copyright claim, then have the wronged party name one item in the offending company's catalog that becomes public domain. Any contractual losses for entities who were benefiting from royalties in the said work are the offender's problem. Not only do the copyfraudsters lose a (possibly much more valuable) item from their catalog, they'd also have to compensate those whose incomes they have affected with their fraudulent acts.

Align the incentives and make the stick painful enough. People being people will take care of the rest.