They should never let it get to the point where they thought that's the only way. The king and one of the most powerful nobles in the nation, and they can't even postpone execution of very important general without making one innocent teenage girl go through horrific execution to scare people? That's gross incompetence and being complacent.
For example, they could make up false leads of innocence, make a spectacle out of general's family and friends begging for his life, to buy some time. They seem to have their own people in various organizations, why is it only Farris that is able to infiltrate successfully and stage various events one after another, while they're only reacting? Even Scarlet's accusation was just a complete coincidence due to her yandere fan, that they ended up using when grasping at straws. Without yandere's assassination attempt on Cecile, or without Lily's later involvement, they would be already at war anyway.
And they're still lucky that people leading Farris and plan to annex didn't go through war anyway through pure spite, when they realized the last one of the Farris bloodline was executed on purpose. Just because Adelbide would be hard to control after war, doesn't mean they would cancel the invasion they have planned with such care, and simply hope they will deal with bigger than planned unrest later. Look at Putin's invasion of Ukraine, historically "it seems like best and maybe last good opportunity to invade the land I wanted, if I don't invade now I might not get a chance later" often outweighed the concerns of "how will we deal with unrest in the annexed territories".