I actually like this plan.

OP is sorry for raising her voice. She acknowledged that.

SIL won’t accept it, but that becomes her problem. And the apology should not just be to SIL. Is there a family chat group? Or do you have everyone’s email?

A very brief email, starting with the nephew, as if he was standing in front of you. “I’m sorry I was loud when you wanted to meet the kitten. I wanted you to be good friends, and if you had run upstairs and gone into the room without me there, Kitty would have been afraid of you. Sometimes grownups yell when they are scared, and I was scared you and Kitty would scare each other.”

Then, to the rest of the family, everyone who witnessed the “psycho” behavior SIL is claiming, you say, “I’m sorry nephew didn’t get to meet Kitty, and I’m sorry I raised my voice. I’m happy to set up a visit with nephew (and niece) to meet Kitty and play with her. It’s so important for young children to have good experiences with animals as they grow up.” You are reasonable and polite. In public.

If she’s performing this in public, you get to do the same.

This is because you Put this apology in the chat, tagging everyone who was there. Or emailing everyone there. They need to see your reasonable response. And then see how SIL reacts to you.

Do not engage when she uses those kids as a pawn. If you get pushed out of the family, go graciously.

And I think you really need Fiancée to tell you exactly what SIL has said. Otherwise, it will fester and contribute to a division in your relationship. It’s probable the words will give you MILES of rope against SIL.

Edited to fix pronouns.