She is NOT a cop thought, she made a deal with him to let him go if he beat her, she is clearly the one calling the shots according to her whims, while a cop wouldn't do such a thing because he'd have clear orders that he must follow.
Also imagine a cop shooting someone down for refusing to join the witness protection program 🥴
Shes the closest thing to a cop they probably have in their world, as shes the literal princess and thus authority figure. So, what exactly is the problem with her calling the shots? Effectively, hes resisting arrest with deadly force. Theres a reason I made that analogy after all.
Princess = the law (and to be clear, no one says the law is
right) telling someone they are under arrest
Sumei = suspect (again, whether he is good or bad, right or wrong is irrelevant) says no
Princess uses force to make arrest. At this time, not lethal force. She broke his weapon, the definition of disarming without injury.
Sumei doesn't surrender. Princess does some more slashing. This is the only arguable bit, since its unknown exactly what shes aiming at with her slashes. It could be fatal attacks, it could be just to wound to disable since hes not surrendering.
Sumei then does his explosions. Also unknown what his aim is here. As the reader, we would be sure hes not aiming to kill, especially since hes the MC, the good guy. But that's exactly what I talked about already; its not a valid arguement. What it looks like is the suspect is now resisting arrest with possibly lethal force. So she responds with increased force too.
The TL;DR is that they both responded to each other's escalation of force, but she's the one that has the authority to do so. You can say shes wrong since Sumei is the good guy, but that's just meta knowledge. If Sumei had been a devil that she attacked, you wouldnt try that argument.