Monkey Majik is an enduring but unlikely success story in Japan’s pop-music scene, with seven full-length albums, a fan base devoted to its upbeat sound and a fistful of hit singles like “Around the World” (which became the theme song for the 2006 TV series “Saiyuki”). J-pop stars such as SMAP, Chemistry and Kumi Koda have sought the band out for its songwriting, which skillfully mingles English and Japanese.

Maynard...