score:7
The Japanese Embassy is correct. Per the Consulate-General of Japan in Vancouver:
Q: I was born in Canada and have dual citizenship, Canadian and Japanese. When I go to Japan, which passport should I use?
A: According to the Immigration Control and Refugee Recognition Act, Japanese citizens are required to use their Japanese passport to enter Japan.
Somewhat oddly, I can't find where in the Act it actually says this though, it may be implied by the other conditions.
My hunch would be that, in practice, the baby would be fine to enter with a US passport as it wouldn't be that easy to connect the dots to her Japanese doppelganger.
However, I'd still get a Japanese passport: they're cheap, fairly fast to process (a week or so), will be handy if she ever needs Japanese ID, and will allow her to enter the country 100% legally. The main fly in the ointment is getting the koseki extract you'll need for the application.