A functioning elected legislature had been in place in Japan for 55 years before the end of WWII. Institutions were in place. Also, the emperor agreed to renounce his divine status. The transition to full democracy was not difficult.
Iraq would be one of two dozen Muslim states. Any government seen to be installed by the Americans would not only be seen as illegitimate from within but, most importantly, from without. There was no other "Japanese state" other than Japan.
The parallels aren't there.
I guess the parallel I was thinking about was where a mortal enemy of our country, in which we were engaged in a war, upon being defeated became a secular democracy, where it had once been a dictatorship. A further parallel that I was alluding to was the nice constitution that we wrote for them. That the two cases are not completely parallel is obvious.