I'm trying really hard to keep suspending disbelief. No matter how rooted societal expectations are, might precedes all. Who is going to stop her?
Surprisingly not that hard to believe. Especially if their nation is a traditionally patriarchy. Even more so if they are bigoted. "Mother" has made several comments already that leads the audience to believe that men are the defenders and bread winners in their society while women cook, clean, and bear the children. Otherwise there would be less resistance to her self improvement....
OTOH there might be other practical explanations for preventing branch family Princess from becoming too strong. If you didn't catch on, her mother is only the 14th concubine to the current Emperor. If the Princess gains too much personal power and authority, she becomes a source of discontent and leverage in political circles.
Not only could she out shine her brother, but other individuals closer to the throne. If so, it becomes less about the proprietary of a woman acting beyond her means, and becomes an issue of politics and succession rights.