Your options if you choose to stay with him is to be okay with him living and working with this lady OR you move in and live with them. You don’t like her, she’s lied about you in the past and you’ll see them be close and chummy. Sounds like if you choose any of these paths you will be very unhappy and the relationship will stay dysfunctional, if not worse.

A 3 year relationship is going to be hard to let go but try and not make it more difficult by trying to prove they are cheating. I don’t think that will help with healing or moving on. The situation he created no longer works for you and thats good enough. Hope you find a better partnership in the future :)