Let me explain you why this scene is really, like really really important, for Edmund. Since he betrayed his brother and sisters in their first time in Narnia, Edmund tried really hard to redeem himself. Not only by becoming King Edmund the Just (The Lion, the Witch and the Wardrobe), but also by supporting Lucy when she said she saw Aslan (Prince Caspian), helping Peter when he fights Miraz (Prince Caspian), joining Caspian looking for the seven lords (Voyage of the Dawn Treader) and much more. He is, and forever, marked by what he did when he was only a kid. A kid mad at the world for making him go away from his mother, from sending his father to the war. I’m not saying that he has to be forgiven, but understood instead.
As it was said, your worst nightmares came alive in there. So when he sees Jadis, the White Witch, he freaks out. I mean, she embodies the past he tries so hard to forget. He has to face, again, what he regrets so much. He is scared. Jadis manipulated him when he was young and naive. And she is trying it again. So, let’s not say that Edmund betrayed his family and he doesn’t care, or he had moved on. Because that’s not true. You can see it in his face, he still struggles with that. He is far from have moved on.
I like to believe that the way the Pevensies said goodbye to Reepicheep shows how their relationship is/was. Edmund always had much respect towards him. Lucy is, and will forever be, fond in animals. She treats Reep like a friend. As for Eustace, Reepicheep was his first friend, and it’s really hard for him to say goodbye.
Remember that little kid at the beginning? That kid who left his brother and sisters for a woman who made him feel important despite the fact he had to make his family his servants? Look at that kid and tell me if you recognize him. Because that kid right there is ready to finish the White Witch for once and all, even if that means he might get hurt. He won’t escape because of fear, because of self preservation. He will fight for his cause, for Narnia, for Aslan, and to help his brother.
Edmund is helping his brother to avoid making the same mistake he did. Being manipulated by the White Witch. Because he cares about Peter. He cares about Caspian. And what makes me sad is that, until then, Peter may not have realized how much Edmund regrets his treason, how he tried to change and redeem himself. And maybe how mad he is at his brother for not seeing that. Yeah, “you [Peter] had it sorted”. Edmund is really trying to show his brother that he can help, he can be a really good ally, not someone to not trust. He is not the little kid he once was.