The Banana Boat Song is a folk song- Harry Belafonte didn't write it, nor is his version an immense revision/reimagining. Much like "House of the Rising Sun," it's a song where most uses are viably public domain, or at least not a huge and expensive licensing issue.
I would bet that it's going to be a musical number, and not a lip-sync like in the movie; I'm curious if they'll justify this in the play, as in the movie it makes sense due to the Maitlands being Belafonte fanatics who listen to his records. (Fun fact: in the original, more plot-centric screenplay, the Maitlands were fans of the Ink Spots instead of Belafonte, and the lip-sync scene was to "Sh-Boom."