A song with the same melody and different lyrics is a derivative work. It does not matter whether the song is to be sung for money, or not. The copyright owner still retains the rights to the melody, and can deny anyone permission to use it in a derivative work.
Furthermore, using the derivative song to make a statement does not restrict or reduce the rights of the copyright holder. Refer to Campbell v. Acuff-Rose Music, Inc., 510 U.S. 569 (1994), below.
See: Derivative work
Derivative works can be created with the permission of the copyright owner or from works in the public domain. (...) The copyright for the derivative work only covers the additions or changes to the original work, not the original itself. The owner of the original work retains control over the work, and in many circumstances can withdraw the license given to someone to create derivative works.
And: Why is parody considered fair use, but satire isn't?
As the Supreme Court explained in Campbell v. Acuff-Rose Music, Inc., “Parody needs to mimic an original to make its point, and so has some claim to use the creation of its victim’s (or collective victims’) imagination, whereas satire can stand on its own two feet and so requires justification for the very act of borrowing.”
Also: Mechanical license
Within copyright law within the United states, such mechanical licenses are compulsory; any party may obtain a license without permission of the license holder by paying a set license fee, that as of 2018, was set at 9.1 cents per composition or 1.75 cents per minute of composition, whichever is more, which are to go to the composition copyright holder. (...) In American law, US Code Title 17, Chapter 1, Section 115(a)(2) states: "A compulsory license includes the privilege of making a musical arrangement of the work to the extent necessary to conform it to the style or manner of interpretation of the performance involved, but the arrangement shall not change the basic melody or fundamental character of the work ..." thus preventing mechanical licenses being used to make substantially derivative works of a piece of music. (...) For example: Puff Daddy wants to sample the opening riff from “Every Breath You Take” by The Police. (...) He is free to hire musicians to reproduce the Police's sound, but he cannot copy from any phonorecord with only a mechanical license.