So, six year olds want to be the center of attention. All of them. So, she feels vulnerable. I think her dad should have 'the talk' and tell her that she can adjust to you at her own pace, no hurry, but she can not be rude. And then back off. Always be friendly and available without pushing. And let her know she's most important to her daddy. I admire how you've been trying to make sure she is comfortable and feels important.
Mixing things with a person who has a child from a previous relationship is hard. It's a big reason why couples end up breaking up. I would forgo having a child with her dad for a good long time as that will just be so hard on her fragile sense of place in the family. And if you are compassionate to her feelings and take this slow, hopefully she will warm up to you!
Oh and please do not be angry for how she is acting. She's not thrilled with the situation. But she's young. The world is still about her. And what she wants is for her parents to be together or for her to have her daddy all to herself. Not unreasonable for a kid to want that. It's understandable. Even if it is not realistic. As time goes on and she sees things are going to be okay and she is still important to her dad, she should soften. AND, the dad needs to make sure she is polite. Don't you do it. Let him be disciplinarian and boss of her and you are behind the scenes working with the dad, if that makes sense. good luck