There may be some (music) publishing company or something involved, but if you've got permission of the writer of the lyrics (who presumably owns copyright or owns the (music) publishing company that owns copyright), you're way ahead of the game. A (book) publisher will work out details, but you might mention in a query letter that you already have permission of the composer to use the lyrics.
I agree that if you have the consent and cooperation of the artist and you are writing something that will promote his work, then you are definitely way ahead of the game. But you shouldn't make any assumptions.
Artists typically do not own their publishing. Generally in fact they're keen to sell the publishing early in their careers because they need the advance and the support a publisher can provide. And then later in their career they see a big publishing deal as a retirement fund. If the publishing is owned by a commercial company then you can't expect them to do anything other than make a commercial decision and the artist's views may not be considered.
I can't remember the details but I know that after Paul McCartney failed to buy the Beatles publishing when Northern Songs went up for sale, he had several disputes with Michael Jackson and Sony over their aggressive use of the Beatles songs in commercials. Having spent a fortune on the rights, Jackson wanted to squeeze them til the pips squeaked.
That said, I would think you will be fine. Just don't spend a year writing without making sure