Both Sakurahime and Shizen Takeru are trying to maintain the peace between the two shrines, and are more often than not working behind the scenes to undermine those within they're ranks who try to provoke a war (hardliner amongst the Shizen branch families and unaffiliated youkai primarily)
Takeru at times issues threats and ultimata to Anmori (especially if a case concerning a missing human isn't solved quickly). The target audience for these are mainly in his own ranks, and Sakurahime is fully aware of this and will generally deescalate the reactions on her own side.
Another tool he uses to deescalate tension is to send an observer to accompany the group sent by Anmori investigate a case. These are usually drawn from people he trusts (including his daughters) and share his dedication to upholding the peace. (You can use this as hook for why a Shizen chara participates)
The path leading to the shrine is considered neutral territory. Even during night, it is more or less completely safe to travel for an unarmed human, and any attack on it will trigger an immediate and overwhelming response by the Anmori Shrine (with observers from Shizen). Almost every youkai in the area is aware of this, and even those that prey upon humans respect it, for they know that their death would follow soon after.
Much in the same way peaceful youkai are considered protected within the town, and especially an attack on those attending school will be met with a quick response by both Anmori and Shizen. Takeru has made it clear that he would consider any attack aiming at the children to be an attack on both his personal honor and the honor of the house, and would not rest until he had brought to justice everyone involved. As a result, even in the most hardline factions within the Shizen branch families generally avoid even the appearance of thinking about such a plot.
The position of the head of the Shizen family is not a democratically elected post, it is an inherited position that goes to the oldest son of the main family. In the case that there are no sons, the oldest son of one of the branch families instead succeeds the position. This means that Shizen Takeru cannot be replaced as head of the house by a vote. The branch families are traditionally expected to be subservient to the main house, so strictly within the rules the branch families have no power to do anything. However, should he appear to be weak, the hardline factions might declare him unfit to lead, gambling that the majority of members will side with them. This could either lead to a hardliner-controlled Shizen or a civil war between those loyal to the ideals of Takeru and the traditionalists, both of which would be horrible for anyone involved.
When a miko of Shizen gets pregnant with the child of a youkai, she is sent to Anmori to give birth safely. While uncommon, a few belonging to the hardliner faction have instead committed suicide instead, as per the old traditions.
Per tradition, a miko of Shizen has the duty to get treated the right to select who treats her. This overrides other rules, including those concerning incest and marriage. While the chosen has the right to refuse, this is very rarely done.
Anmori handles treatment informally and generally leaves it up to the miko to decide if and by whom she wants to be treated (though there are a few cases where Akane and/or Sakurahime orchestrated things in the background so that two people who had interest in each other (and who they thought would make a nice couple) found together (in their opinions, the chance to find love as result of a tragedy both helps to heal the wounds and is good in itself)). Of course, as Sakurahime can make the pregnancies safe, there are also a few miko who decide to go untreated and give birth to the hanyou instead.