how did the MPAA not file a lawsuit againt this movie?

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The secrecy of the MPAA was not necessarily warranted. They are not like a company that may have proprietary secrets or protected work that would damage them financially if revealed. This is a group whose work impacts a large number of people.

They would argue that if the raters were not anonymous they would be open to influence by the movie industry. Since it's the movie industry that created and maintains the MPAA that's a specious argument. If one of the raters were to take a bribe then that's a personnel issue.

Meant to be kept secret by the MPAA, though. They didn't do anything illegal in the film, so there's no lawsuit that can be filed.