Inversion/non-inversion in wh-questions with long phrases after the wh-words
The basic structure is "I don't know [who the first man [that ... ] was]"
But the long subject inside the subordinate clause may trigger extraposition, whereby it is moved behind the short VP 'was':
I don't know who was the first man that ... .
However, I would use a different extraposition, and move only the embedded relative clause:
I don't know who the first man was that ... .