-[X] There's a chance to break that cycle now - but only if both parties agree to lay down their arms and make peace. If Rei can do what you could not and walk away from her revenge, if Ritsuko can accept that as proof that Rei is more in control of herself than Misato and is less deserving of death than you are.
[X] "What the fuck's wrong with me!? What the fuck's wrong with you?! You are trying to murder a child, and that child is asking not to be murdered. How did you get like this, Rits? Executing children, talking like a cultist? We went to university together! If it wasn't for me, Rei would already have gunned you down. You're welcome, by the way."
-[X] "Now, I want answers. What's this business about Rei's soul? What's that thing about a donor for Unit-02? What else am I being kept in the dark about!?"
-[X] Tell Rei that you, too, are intimately familiar with the feeling of wanting justice for dead predecessors - your father's demise was the very reason you took up arms and joined NERV. Killing the Angels was your way of attaining it. But look where that has gotten you - you are an alcohol addict who marches children into battle, landed two of them in hospital and nearly got killed at the start of the last Angel attack. From someone who walks that path, you can attest to Rei that if she wants to live and keep the memories of her predecessors alive with her, starting down that road of revenge by pulling the trigger will not be conducive to a happy ending.
-[X] Remind Ritsuko that part of the reason Rei is pointing a gun at her now is as much for revenge as it is out of fear. If Ritsuko were to kill the current Rei, what would stop the next incarnation from eventually developing the same feelings about her predecessors' fates? Ritsuko would have to kill that one too, to stop her from taking revenge. And the next, when it happens again. And the next. You wonder if that wasn't the reason a number of the earlier Rei clones were killed to begin with.
-[X] If Rei shoots, that means she dies. That means Misato dies. That likely means Shinji, Asuka, Hikari, Toji and the whole of humanity dies. You're not going to tell her to let go of her hatred - what NERV has done to her is beyond criminal - but she has to weigh up her courses of action. Is blind vengeance worth that? Is it worth failing to keep the memories of the other Reis Ayanami alive?
-[X] You're defending a child who has been murdered over and over from someone who's trying to kill her, Ritsuko. Try and be actually objective for once - examine her behaviour here. She's trying to argue why she should be allowed to murder a child while that child is asking not to be murdered. What the fuck is wrong with her? Does she hate Rei that much?
-[X] Tell Rei that you, too, are intimately familiar with the feeling of wanting justice for dead predecessors - your father's demise was the very reason you took up arms and joined NERV. Killing the Angels was your way of attaining it, and you were afraid there was no other way. Now, you've become someone who sends the children they love into battle, too blinded by vengeance to wonder if you were doing more harm than good. From someone who walked that path, you can attest to Rei that if she wants to live and keep the memories of her predecessors alive with her, she needs to take a risk and trust that others will see and value her humanity if she opens up and shows she's not dangerous to them.
-[X] Ask Ritsuko what she means when she says that Rei "could lead to the extinction of mankind". If she is going to be treated like a menace to be kept under control, it's no wonder she feels like violence is the only option. If Rei doesn't feel like she can trust anyone, and you don't feel like you can trust Rei, there's no reason to think the same thing won't happen with the next Rei. And if Rei is somehow "dangerous" in a way that goes beyond merely being an Eva pilot, it's even more important for her to feel like she can trust us. Because she certainly can't trust Gendo, but she might be able to trust us. There has to be a better way then just recreating the same problem over and over again by killing her.
-[X] Rei, just like you, Ritsuko also deserves to feel safe which she cannot while you point a loaded gun at her - put down the gun first and then I will do so and collect both.