Sansui never gave his wooden sword for Brauer to hold. He didnāt bother even āunsheathingā it for almost the entire fight.
He does the fight a bit differently. First, fighting her normally. Then trying to teach her things like patience and whatnot. Eventually it gets to the point that it is so one-sided that despite Rin not quite giving up yet, he still easily handles her. With Rin on the ground due to all the damage she took, he finally āunsheathesā his wooden sword and explains that if it wasnāt for the āno killingā order, he bluntly says loudly that her being a berserker is a risk he believes is too dangerous to leave alive, and he has no qualms killing right her here and now. That his only regret he didnāt realize this sooner and didnāt take care of the problem before this point. Since this fight was so one-sided, Rin realizes immediately that she will die and breaks down crying.
Sansui achieving the goal of breaking her spirit, does nothing and leaves her be. This brings up a problem for the noble family Sansui is serving. They never had an instance where Sansui would be against an order theyāve given and believe itās high time to change his position of bodyguard to something else. Since they believe that if they gave another order that countered his will, they believe that Sansui would most likely ignore it, if he believed it was for the best. So they plan to have Sansui become an instructor instead and eventually sooner to marry Brauer.
The only reason why there was a āno killingā order was because the Kingdom wanted Rin alive to teach other berserkers in order to give the kingdom more chances at rare arts and stuff. Since they lack an actual Trump card. The order was pushed through the noble family Sansui is serving. The Kingdom annoyingly is so obsessed with strengthening themselves because the Big four noble families each has a Trump card. And they still havenāt gotten over it.