As a physician who works with radioactive materials and patients every day, I can assure you there is no reason to avoid Tokyo or its airspace. The additional exposure you might get from going through Japanese airspace is so incredibly minimal it will have no significant effect on your health.

This whole fear of radiation is somewhat overblown by the media. Air everywhere contains radioactive particles, most naturally occurring, as does the ground you stand on, the food you eat, and your body itself. You're constantly exposed to radiation, and your cells are designed to cope with it and repair themselves.

Statistically speaking, your risk of being murdered by a co-worker is 50% higher than the risk of death from living next to a nuclear power plant for 30 years.