In a way the fact that he ignored Sei was a boon for her. It meant her first emotions were anger instead of despair, which is a more "driving" emotion and her first desire was to get the hell out of the sphere of influence of the prince, which lead her to seek out other people and learn about her new environment on her own terms, making some real friends along the way.
Aira's pov really brings that home.
Of course the fact that Sei was an independent working woman helps too, but if her reception had been different and more welcoming, I'm not sure she would have faired as well as she did. At the very least it might have taken her longer.