Here is how I understood the show.
Brother Day has absolute authority. He is Empire and the emperor, and that is it. Dawn and Dusk have authority through Day, but on their own have no power. He could end the genetic dynasty or disband the empire tomorrow.
Demerzel is a robotic entity. She functions on her own, and the laws of the empire don't have any magical effect on her.
In her core programming she has 1 directive, protect the Empire.
So what happens?
In really there is no system in place for Dawn and Dusk to exercise power. Dawn may be a child, but Dusk is an ex emperor with huge political say. They are supposed to be the same person and always agree on everything. The people must see the three of them always agree on everything. In that moment in the throne room things go to chaos. Day and Dusk start a fist fight. If anyone seems them that could be the end of Empire. Demerzel does what is needed to end the fight and save what she must protect, she kills Dawn. It's like taking the toy away from two children who have never learned conflict resolution.
She can do what she does because it is for the good of Empire.
After she does it there is no replications, because Day knows what she did was right. He knows he lost control of the situation, and she saved him without him backing down.
Unrelated I think she is the one responsible for the genetic drift. The end of the clone dynasty is for the good of Empire.
We also know the laws of robotics don't apply to her, and most likely to anything in this universe. We have seen her kill on orders, or her own will