Cegnore was an isolated environment, but rumors still reached it. Many weren't privy about Neuria's recent events, but almost everyone knew that Khan had scored a spot in a political team featuring an Ambassador.
Rumors about other Captains would have struggled to echo that far, but Khan was unique, and he knew that. His fame gave value and weight to his claims and requests. Winston would have never taken him seriously without actual proof otherwise.
Khan's awareness of his status had given birth to his current attitude and behavior. That wasn't only the result of his fame, and his approach wasn't random either. He was choosing his words to get what he wanted as fast as possible.
The Nak also played a role in that choice. When Khan's main goal was involved, he found it easier to show a firmer and darker stance. He couldn't bother resorting to compromises and nice words when more straightforward options were available.