The challenge is to make Japan a Muslim-majority state. Bonus if it's a secular one.
Interesting challenge. I'm not convinced there's something unique about Japanese religion that makes it immune to Abrahamic monotheistic religions (see the nascent spread of Christianity for example).
You need a couple of things:
(1) An impetus for Islam.
(2) Introduction of Islam either prior to Toyotomi Hideyoshi unifying Japan (mid 1500s or so) or somehow, an opening up of Japan prior to the American expedition to the islands.
According to Wiki, there was an Arab on the initial Portuguese expeditions from Malacca that preached Islam to the Japanese, but we'd probably want something a bit more widespread than that.
You could have an entity like the Malacca Sultanate somehow make contact with the Japanese via the Wokou pirates in the 15th century and begin trading with the various fractured states. Just like in Southeast Asia, Islam could spread through some sort of trading environment.
That's one potential idea and the butterflies would be enormous for any potential unification of Japan.