I'm going to tell my ex that I won't get back with them unless they cut ties with a mutual friend
Why do you need to do this to your ex?
I really don't like that mutual friend anymore.
You should not be telling anyone who they can or cannot hang out with. It is none of your business who your ex chooses as a friend. You need to stop the controlling behavior or your ex will not want to get back together with you.
I am a manipulative asshole and I don't like others getting close to my ex so I will manipulate them into it. I am attracted to them.
Giving your relationship partner ultimatums is a bad idea. You should want to be with your partner to make you and her happy. If you have to make ultimatums than maybe your relationship is not really worth being in.