I'm going to let my cat run away so I can contact my ex
That's an odd way to do it. Why do you want to contact your ex?
I miss my ex and I want to hang out with my ex.
It is normal to have feelings for your ex, but you should not use deceit in letting your cat runaway so you can be with your ex again. Your pet will become lost which is bad thus your ex will blame you for it.
My ex cheated and left me for someone else. They loved my cat so I want to use it to get back into contact
I think it's more wise to just fully cut them out of your life since they cheated on you. I understand feelings linger, but don't you think it's preventing you from finding someone who truly loves you? How about trying some dating apps and getting back into the scene?