[Platapus]

I don't know what the legality would be for Japan to shoot down a missile that is not in their sovereign airspace.

Anything above the Karman Line (100km) is recognized as international space (no pun intended)

Anything under 30km is considered territorial airspace. The area between 30km and 100 km has not been decided on an international level.

The primary reason is that very little happens in the 30-100km as it is too high for aircraft and too low for space craft. The only things that go though this area are space objects going up or down.

The Korean missile was passing over Japanese zones at about 500km, well above the Karman line.

The THAAD has an intercept altitude limit of about 150km so there was really no way the Japanese could shoot down the North Korean Missile as it was passing. I don't believe that the Japanese have an interceptor that has a higher operational altitude. Also these types of anti-ballistic missile interceptors intercept on reentry or the downward path.