What should I do if I need to talk to my ex about how she represents me to our kids?
That seems like a good reason to have a conversation. Is there something she does specifically?
She always talks to them as if I'm a bad man and they should be afraid of going home with me
That's a pretty rude thing for your ex to do. Wanting her to stop badmouthing you in front of your kids is totally normal and I encourage you to do it. No child should feel afraid of a parent.
Nah, you don't get it. If I tell her to stop she will make our children's lives a living hell.
It's good that you want to stop her from badmouthing you. Hearing negative things about you is hurtful for them. It's best that you talk to her about how you feel without the kids present so that she feels less defensive.