] The other day I was watching a documentary about the sinking of the IJN bbattleship Yamato.

IIRC, the Yamato took fewer bombs and torps to sink than her sister, the Musashi at the Battle of Leyte.

] did the RAF or Commonwealth ever operate heavy bombers over the Japanese home islands?

Don’t know, but IMO, they were probably busy recovering Hong Kong, Indonesia, Malaysia, Singapore, &c, in the SEA.
After WWII, ISTR they occupied part of western Japan, but withdrew after a few years, unlike the Yanks who are still all over there.

FWIW, during my research on military aircraft accidents in Japan, from after WWII to now, I’ve read only one accident:

14 January 1946:
C-47.
RAF.
Emergency landed, at Kitatatsushima, Sado City, Niigata Prefecture.
After 40 days, repaired and took off.

http://www.ne.jp/asahi/aikokuki/aikokuki-top/NAH/ChronicleShouwa2.html