Ten is a hard age to adjust to strangers becoming family. While you may not be a stranger to her anymore I could understand her having a hard time accepting new family members and lashing out.
I would strongly suggest counseling for her and counseling for you and your fiance if you can't be on the same page as him. Merging two families as one is a difficult process for everyone.
My experience is my dad left my mom for another women who had four other children when I was roughly the same age as your step-daughter. She's only ten and is not mature enough to handle this kind of change in her life without help.
I would try to think of her as your daughter too. Kind of try to bond with her. Then as your partner sees you have her best interest at heart, he may feel more open to letting you help set rules on behavior and such. If you and he are parenting her together with her mother, he may not see it as someone trying to put her down or be harsh with her every time you want to correct something. Are you two having a baby together? her dad and you?
I would nanny cam it. Show him how his daughters acts when daddy's not around. And how dose she act toward your daughter?