score:7
There's no law requiring you to "use" a US passport when you leave the US. The statute you link to requires only that you "bear" a valid US passport. If you have it with you, you are certainly bearing it. Furthermore, there is no penalty for violating that law; the penalty was removed in 1978.
I check in all the time for flights departing the US with my non-US passport. Nobody has ever asked about my immigration status. If they did I would simply show my US passport. If they asked me why I didn't check in with it, I would say that I'm going to use the other passport at my destination and I don't want to get the airline or myself in trouble by showing a different document from the one on the passenger manifest.
Upvote:0
Regarding Vietnam, to save pages, at least as a US citizen (what is your other citizenship?) you can simply get an e-visa instead of an authorisation to get a visa on arrival. You will then only get entry and exit stamps in the passport.
Also, airlines don't usually care about your immigration status, and as you said the US doesn't do border control on exit. So when checking in for the flight to Japan, you can show either passport and it won't make any difference regarding the space in your US passport.