Since the Rimbault broadside - if it exists - gives the same text, Smith could just as easily be the author, no matter when the sheet was published. The broadside might have been reprinted from the book, with or without Smith's name on it.

That seems most likely to me, but there is simply no way to know.

If the broadside was reprinted from the book (already accounted for in 1656), would the Stationers Register be expected to have an entry for it?