I just started playing this game, I'm not familiar with the plot, I don't know if I understand your plot design correctly.Well, if there the icon is set, there must be a reason for this and players missed to trigger the scene. I soon will look at this and look what happened.
But I think the control variable for Kate's storyline is "sp3var", when the storyline progresses to the point where Kate invites the MC to her live room, the variable is sp3var = 5, then you can set a variable sp3var = 17, and your Kate storyline will only go to 16.
At the same time, you can change the if condition in the quest hints to this one, so that it won't affect the quest hints, and also remove the PC icon false hints.
When the plot proceeds to the MC's hand injury, you can change the variable back to sp3var = 5, and the PC icon prompts for that quest come back, I've done a simple test on my own to make it work, and it doesn't affect the PC icon prompts for other quests either.