What I have heard is that if you renounce your Japanese citizenship, you have to turn in your Japanese passport but if you tell Japan that you are renouncing your American citizenship there are no checks to see if it has been done...
I do not know how true it is and I would not recommend breaking the law.
There is also the matter of detection; Japan is very strict on watch entry and exit, any use of a non-Japanese passport would be noticed there but is there any way for them to see if you have a second passport beyond that?