@Goldbracelets - nor surprised you haven't been back to the thread for a while!
Look, she's 13. It's a PITA age at the best of times, and this isn't one of those. She obviously feels fiercely loyal to her mum, and regardless of what her mum is saying about you, she's probably also hoping that her parents will get back together. Even if she knows that isn't ever going to happen, your existence in her father's life does definitely exclude that as a possibility.
Another point to consider is that she doesn't want to be put in the "piggy in the middle" position by her mother - who will definitely want Details of you, so she can rake over them.
So what can you do - I would leave her alone, in all honesty. Do NOT ambush her - that will create such an almighty mountain of resentment there would be no coming back from it.
Your mother might have a professional view on the situation but it's not necessarily helpful when you have a recalcitrant 13yo on the other end of it, who doesn't want to get caught up in what she sees as her parents' drama.
I would suggest that you tell your fiancé that he needs to ASK his daughter if she's ok with you getting engaged, not just tell her it's already happened - if she says no, then that's too bad because it's going to go ahead anyway, but it's marginally better than telling her flat out it's already a done deal.
I have a friend whose DH had DC from a previous relationship. The mother did the parental alienation thing - he would go to collect, the DC wouldn't be there, or she would refuse to answer the door. As a result, his DC (now adult) and my friend have never met. They've been together for well over a decade now.
Obviously your situation is different because your fiancé's DD DOES stay with him a fair bit - but you will, at some point, move in together and she does need to come to terms with that.
Your fiancé needs to sit down with her and give her a rough timeline of when you expect things to happen, so that she can mentally adjust to the situation. She cannot expect you not to be there when she comes to stay, if you're living together!
I feel my post is a bit all over the place but take home point is that your fiancé needs to have a chat with his DD, explain that things will be moving forward with you, give a rough timeline to give her the chance to come to terms with it all, and then leave her to absorb it for a while. She does need to understand that you will be there and she can't avoid you forever, if she's coming to stay with her dad regularly.