You can get a British passport quite easily for your child in Japan. See Mike's post re the Japanese document required for proof of birth.
What I want to warn you about is the difference between registering your child"s birth with the British Government while overseas and doing so while in the U.K.
This may have changed, but it is very important for your child (and future Grandchildren) that you get this confirmed with the British Embassy in Japan.
If your child is registered as British while overseas, they do not automatically have the ability to pass on British nationality to their children. Where as if they are registered within the U.K. they do have this right.
My children were born in Japan and this was spelled out to us emphatically when we sat down with one of the officers at the Embassy in Tokyo. Following this advice, we travelled to the U.K. with my children using Japanese passports, and registered them there.
They are adult now, have their own British passports and transferable nationality for their children.
This was more than 20 years ago, so you will need to check on current legislation, as it is particularly complicated with the British / Commonwealth system. Give the Embassy a call and arrange an appointment to sit down and get it all officially explained and confirmed. If it is still the same, your Grandchildren may thank you for it one day.