I don't know why she's stubborn, but it's not pride. She bows down to others too easily anyway. She says it would be disadvantageous to ask beyang for help, but beyang's not interested in demanding anything. I'm thinking it's along the lines of them firstly not helping with the dragon slaying alliance, and secondly that compounding with owing a favor for helping now. Crow's debt to beyang would be absurd at that point. It's weird since she know Mu is just, soooo...
We will never know why she's actually so stubborn, unless at the end of the arc her admiral will ask why.