Perospero, Big Mom and their family are RESPONSIBLE for Kidnapping Sanji and threatenning to kill Sanji's family.
Besides, Carrot being mad at Perospero specifically for that is dumb since he had zero involvement in it. Bege was responsible for capturing Sanji while Linlin gave the order. She might as well have a personal vendetta with Cracker, Snack, Smoothie and the other 80+ Charlotte family if she is beefing with Pero over that.
You can't kidnap and threaten to kill someone and not be taken responsible when the friends of said someone comes into your territory to beat you up.
Peropero had only ONE thing to do : Let the strawhat leave. His chooses to prevent them from leaving, he is therefore responsible for the choice of Pedro to blow himself up.
Allowing individuals that are clearly hostile to his beloved homeland and that have proven to be able to exploit weaknesses in the nation's security arrangements, weaknesses that they could share with multiple powerful hostile factions, would be clearly detrimental to his nation's interests.
Sir Perospero rightfully chose to follow his duties to his nation, Pedro rightfully chose to follow his personal obligations to his new comrades and Carrot unceremoniously chose to irrationally whine about the result of the choices those two men made like a little 🅱️.
Katakuri understood this, this is why he let Luffy go.
Sir Perospero should be elevated to Sweet Commander for his loyal service to the great nation of Tottoland.
Say it with me folks!
Perospero did nothing wrong!