Have an invasion launched by Kublai Khan or one of his successors partially successful, so that you get a large section of Japan under a Mongol/Chinese dynasty & to some extent sinified but a also more purely 'Japanese' remnant-state at the other end of the islands?
I wrote a short TL once, elsewhere (Do any of you already know the gaming apaazine 'A&E'?), that had a successful invasion under Kubiai Khan leading to subsequent conquest by the Ming, early in that dynasty's time of power, and then a Ming survival there into modern times after the Manchu conquered mainland China. The later Ming ITTL were a bit more open to Western influence than either the Manchu or OTL 'Shogunate' Japan, and although their modernisation was slower than OTL 'Meiji'-era Japan's they'd reached a comparable point by c.1910AD. There was an influential 'Church of Japan' (bascially Anglican style, but with reverence for ancestors accepted and some other 'local' features) to which the emperors had converted and whose spread they were strongly encouraging, and then during the 1920s -- after the Manchu had collapsed -- they launched an armed "return" to the mainland.