I hope the following helps a little
I attended an international school in Japan for 1 year before moving to England at 6, I have friends from said school who were there for 3~6 years before moving to England (apparently I started a trend!). We all found the transition to an English country easy, as our English was about as good as any of the locals. Those that were there for a longer period of time had fluent Japanese (perhaps because they were actually in the country). Being in the country definitely makes a difference, as I once heard, immersion is much better to learn a language than any school.
However, personally my Japanese had plumetted after moving. I went to the West Acton saturday school to keep my Japanese, I attended the class for natives (if I remember correctly there were also classes for those born and raised in England), but I was left behind and my motivation quickly disappeared, my sister and I quit the school after 3 years. I guess for your children it would also depend on wether they actually have the motivation to keep and/or improve thier English abilities.
Now on bullying, as in any country bullying happens, and as in any country it affects some more than others. In an international school, because we were around so many of our own, racially-motivated bullying was non-existant, but ofcourse it was a school, so bullying for childish reasons still existed (height, stature, etc) but nothing serious. As for normal Japanese schools, as Japanese society outside of Tokyo goes, don't be surprised if they are singled out in both good and bad ways, but in both cases it really depends on your children's personalities and attitudes, if they can confront it head on and use the fact that they are "half" to their advantage, I'm sure their time there will be easier.
Back to language (as I'm sure this is important to you), since leaving the school my interest in Japan shot through the roof, and I went on to learn the language myself, I beleive I have a good knowledge of the language (those on the Japanese section of this forum have been kind enough to agree) but my speaking still has a pretty bad accent to it, and my kanji writing is bad, without a computer that is, haha. My sister has not shown an interest and has forgotten most of her Japanese (except for basics, but she still talks with the vocabulary of a 7 year old...). Even though I speak the language at home it really doesn't cover a broad range of subjects ("I'm making --- for dinner" "OK") so I wouldn't really count it as practical practice. Perhaps making English-speaking friends will help keep your kids' English better than just speaking it at home. In England, since my friends from Japan moved a long while after I did, and the few I do keep in touch with I speak to in English, I had no friends who spoke Japanese.
Hope that helps a bit! (Whew, sure was long enough!)