Traveling with 2 passports , one invalid?

This is my situation:
i need to leave US on the 7th of September to Romania. I dont have my valid American passport because I needed to send it to apply for a British visa but still haven’t gotten it back yet due to some IT problem at the UK consulate.
But now my question is would I be able to leave the US with my Vietnamese passport (I have dual citizenship) and enter Romania with an invalid American passport (not because it’s expired but because it’s reported stolen but then I recover it). The problem is Romania won’t allow me entering the country with my Vietnamese passport without an invitation from an authority even if i’d only be staying there for 10 days .
Should I show the Romanian customs that I have an American passport and not the vietnamese passport hoping they would let me enter the country
Sorry it’s a bit confusing.