Even in the lawsuit they only claim copyright on the “song book” and not individual songs
The lawsuit does say:
Watch Tower is the author and copyright owner of the musical composition “Listen, Obey and Be Blessed” (the “Composition”).
Watch Tower also is the author of and owner of several other copyright registrations
for original works of authorship concerning or embodying the Composition, including: (a) U.S.
Copyright Registration SR 864-110 for a sound recording entitled “‘Sing Out Joyfully’ to Jehovah:
Song 089, Listen, Obey and Be Blessed,” effective as of December 30, 2019