You're in a tough spot, but let's be clear first: inviting just your new guy without her is way worse. It'll look weird, hurtful, and you're gonna create huge awkwardness. If you don't want her there, own it and say it straight up, not in some roundabout way. It's more honest for everyone.
Now, do you really have to exclude her? Three years is still a serious relationship, and if you have mutual friends, she's gonna find out that everyone's going except her. That's gonna be tense. If she's moved on (since she's with someone) and you guys can be cool together at an event, honestly that's the smoothest solution. You'll just kind of ignore each other, no big deal.
If you really can't stand the idea of her being there, talk to your fiancée first, then reach out to your ex like an adult. Like a simple message: "listen, for the wedding we decided to keep it to family and close friends, I hope you understand". It's not a crime to leave someone off the list, but it's better if you own it rather than play games. And mutual friends will get it better than if you were picking and choosing randomly.