I agree her name should be Trang Thi Thu Nguyen.
My question is at what point do you want to start correcting her name? I heard stories where you can have the fiancée’s name corrected when getting the marriage certificate.
Then when applying for GC, just show the marriage certificate and have the name corrected there as well. From that point on, her name will be the correct one on every other legal document.
Not sure about SSN though. Might have to match the passport still.