As you appear to have used your dd's true christian name twice I would suggest you report your post and ask mumsnet to remove any reference to her name, OP.
When saying that your baby's dad treats your dd 'differently', for the purposes of this reply I will assume that he is not as good/kind/patient with her as he is with his own child - please correct me if I'm wrong as it may be that you are implying that he treats her differently than her own father who, while he may no longer be in the picture, is theoretically and in every other way the man who fathered her.
By your own admission, this little girl has witnessed things that no child should see and has been through 'so much' that no adult should go through, let alone a 7yo.
You say that 'after the shit we found ourselves in a new relationship for two years' at the end of which you gave birth to your ds, but what you are actually saying is that, after all the dysfunctional tumult of her first 4 or 5 years, YOU, and you alone, embarked on another relationship with a man who she was expected to call 'dad' and now he's no longer living with you.
Again, you say that you've 'always been able to talk to her' and that you 'don't lie to her'. This is concerning as it sounds as if you regard your little dd as being your friend and confidant rather than the daughter and child she is and this is also demonstrated by you claiming that she has 'become a cow', back chatting and refusing to listen etc. It's as if you're talking about another adult instead of a little girl.
At least you've recognised that her seeming ability to adapt to change 'extremely well' was in fact an ability to mask her inner feelings by suppressing them - quite possibly because the two men she's called dad who have been in her life so far were, in her perception, scary creatures who she didn't dare offend.
You say she has been 'resilient', but it seems to me she had no choice. In essence, your dd is further proof of Darwn's theory that the species must adapt if it wishes to survive and it's heartbreaking to think of her learning at such an early age that she had to mask her emotions and be a polite and compliant child in order to gain approval.
What chance did she have of being anything else? When has she been able to be a child, living a happy and contented life in a warm and loving family without the cares of the world on her extremely young shoulders?
Regardless of whether her latest 'dad' has been kind or unkind to her, now that he's gone she's acting out. She may feel confused and angry with you. She could be testing your boundaries to find out if you'll continue be the constant adult in her life. Or she may be mirroring some of the adult behaviour that she has witnessed.
Whatever the reason(s), your dd is unhappy and insecure and it's a credit to her that she has not displayed any jealousy of the new arrival. In this respect she sounds as if she's a very caring little girl which gives me confidence that, with careful nurturing, she may be able to overcome some of the damage that's been done to her.
As to what you can do, praise her lavishly for good behaviour and, unless it's adversely impacting on other children, ignore bad behaviour. If she trips, falls, etc, pick her up and give her a cuddle. Talk to her about when she was a baby and how all the things you do for your ds are what you did for her. Read to her; not just at bedtime but anytime baby is napping or you're able to cuddle up on the sofa with her.
Watch films/tv together and talk about what you've seen. Tell her constantly how much you love her and how proud you are of her. Establish a continual dialogue with her - she's crying out for your attention and you must give it to her no matter how lacking in energy you may feel.
Baking together shouldn't be a 'treat'. Involve her in cooking and give her small tasks to do, even if it's only handing you the ingredients. Give her a set of simple chores such as laying the table, keeping her room tidy and putting her dirty clothing in the linen bin/washing machine. Make it sound fun rather than onerous tasks that she must complete. Praise her when she does them.
Give her some crayons/colouring pencils and plain paper and ask her to draw you a picture of your home and all who live in it and another of you, her, and the baby. A child's artwork can be be a clue to the workings of their inner mind and you may be able to discern where she ranks herself in relation to you and baby ds.
Buy the book 'How to talk so little kids will listen and how to listen when little kids talk'. You'll find cheap copies on Amazon or eBay and you'll be able to use the techniques when your ds grows.
Talk to your dd's school. Be as honest as possible about what she's gone through, ask them to observe her and make a referral to a therapist if they think it's necessary. Approach your ds's health visitor for advice on coping with dd and a baby. Consider counselling for yourself so that you can come to terms with the trauma you've gone through and the effect it's had on your dd.
Raising two children takes twice as much energy as raising one. Get as much as rest as you can when dd's at school so you can replenish your batteries ready for the onslaught when she comes home.
Some of the above may seem harsh, but imo it's vital that you draw a clear boundary line between you as the parent and your dd as the child and recognise that your dd's current behaviour is a direct consequence of your own.
Don't be impatient with her; she needs all the love and care you can give her and that means focusing your attention first and foremost on her in order to heal some of the trauma she's undergone.
If you haven't done so already, please enrol on the Freedom Programme and fgs don't move another man into your home unless he's an exceptionally kind millionaire who's willing to look on your dc as if they were his own and has asked for your hand in marriage and, even then, get the ring on your finger before his feet cross your threshold
I didn't set out to write such a long post but I knew that if I cut it short it wouldn't be my only post on your thread. Even so, depending on what further information you provide I may be back how Schwarzenegger does that sound?