CatFlow

Why don't you get a passport from a Vietnam Embassy/consulate in the US?

Getting both passports is the best he can do. He will have to carry both passports when traveling.Show US passport when boarding the plane, then show UK passport when arriving at border control once in London airport. The opposite when leaving UK towards the US.

The UK passport will make it easier for him to travel within Europe, and other countries such as Brazil, for example (US citizens need a visa while UK citizens not).

You were inspected even if it was just to the border area. Just explain that you don't have an I-94 because you were admitted with a BCC.

Did the child enter the US before his mom took the oath of citizenship?

If he did, the child has been a US citizen since the day his mother took the oath of citizenship. All you have to do is apply for a Certificate of Citizenship to get him a US passport.