And historically, no known society has been able to pull out of such a severe dowanward fertility spiral. None.
In a couple of generations, in a "One-Child-Glorious" society, each and every child has 2 parents, 4 grandparents, and 8 great-grandparents all to him or herself. And no brothers, sisters, uncles, aunts, or cousins.
Do you think any such child, with eight great-grandparents to care for (either directly, or through taxes) is going to decide in their 20's or 30's to get married and have kids?
No, Japan will not "stabilize." Nations which will not reproduce, are replaced by those who do.
Japan is an island. They can keep out “invaders”. I wont be surprised to see them going nuclear in a decade or two to protect their territorial integrity.