I think so far what you are doing is pretty much on the mark. Keep speaking with her , keep correctin her , keep showing you care for her. Its obvious its for attention and at that age , they tend to pick up nasty habits from their friends too.
My SD is almost 13, she is derrogatory towards others. SHe tends to be nasty to place herself in a higher position. Something she has learned from her mom unfortunately. We've always corrected her and we got the usual answers ' i don't care if someone does it to me, ' But we did the opposite. Her father told her, well if you dont care, then you are the one who loses out in the long run. No one else. You put yourself in that situation, you did it to yourself, then dont cry about it. I love you but if you dotn want to help yourself, then stay put and have not friends. She didnt' like that but thought twice before saying i dotn care anymore.:)
I can't give too much advise on this. We're in a similar situation. Ive been with my hubby for 6 years and have a 29 month old. His two kids visit EOW.So we dotn raise them. I would say % wise its 90 % at her moms and 10% at our house. If they decide to come that weekend. My hubby doesn't believe in forcing them to come. And lately the SD has shown her true colors. I get along with her but for the time she is here. I see how she is when she is out of our company and frankly i'm glad i do not have to deal with her on a 24 hr basis. Her own brother thinks she's nasty and admitted last week. My DH was in shock but then was really disappointed in his daughter and then anger set in. I told him it is not his fault. We haven't raised her with those values and hopefully as she grows up she'll iron out. An expression we use in my culture. At 9 years old kids do not really see the reality of life. They really do not understand what is going around them.
From what i 've read in your post, it sounds like your DH and you are on the same team and that is great! Her behaviour is normal but keep open communication and ask her why she feels stupid? did someone at school say something? Deos she feel left out and less important. Having a younger child sometimes makes the older ones revert to get attention. Even negative attention. Its normal. Spend more time with her in those cases. Make her she has a spot in the family pie.
As well, she spends 60% of her time over at BM place. Is something happening that is getting her angry?
My SD last month said that she 'hated her mom' Then last week she was teaming up with her against 'our house' lol...its a tennis game with them unfortunately.

We've had (somewhat) similar bad behavior. A__ has no siblings to fight with, but he tries that sort of thing with us... (Tells his dad he hates him, blames his bad behavior on the cat, calls his mom an idiot, etc)
Our approach isn't working like a charm, but it's improving things... He got sent for a timeout every time he said something really mean (usually something deliberately sexist to me, or "I hate you" to his dad)... If it was something sexist or something else that he said to be mean, but may not understand ~why~ it's mean, BF would discuss it with him after a few minutes of time-out. If it was telling BF that he hated him, BF would tell him something along the lines of "Families love each other. You can come out when you're ready to say I love you instead". He'd be in the other room for anywhere between 2 minutes and 2 hours.
That took about 2 months of a lot of timeouts (probably one a day for the times he was with us) but I expect it would have taken a lot less if we had him more. But he hasn't said anything to that degree in a few months now, and for "lesser" mean things, we just respond right away with "That was a mean thing to say" and ask him to apologize or think of a nicer way to say it.

When he has "bad days" though, he can be a nightmare... He was having a bad day last week when we picked him up for soccer. When BF went in to Grandma's house to get him, he wouldn't come away from the TV, screamed and cried (I heard him from the street), and threw a remote. He wouldn't talk to us in the car, except to tell us not to talk about him. He ran away in the parking lot and again once we got into the soccer centre... Just an awful day.
On the way home, A__ was trying to tell us something about school and we didn't understand what he was talking about, so we kept asking questions to try and figure it out. He screamed "You're an idiot!!" at BF and punched the roof of my car. We were on a side street, so I slammed on my brakes, pulled over (it was surprisingly dramatic), turned around in my seat and said (quite forcefully, but not yelling) "Your dad is not an idiot and does not deserve to be called one. I don't ever want to hear you call anyone that for asking questions again. And if you ever do something like that to my car again, you won't ride in it again for a month."
I don't usually respond like that, so it shocked the bejesus out of him and he immediately apologized to both of us and was good for the rest of the evening, and even finished telling us about the thing at school in a clear and concise manner.
It's not that I would recommend ripping the car to the side of the road to scold your stepchild... But in this case, BF's attempts at being calm and using reason weren't working, and were appearing to make things worse.

Another thing that has worked with A__ when he's not being very nice, or not listening, or whatever... Is to remind him of a recent time when he was listening or being nice and tell him "See? We know you can be really nice and good. Remember how much fun we were having at that time? Wasn't that better than now, when you're being mean(or not listening, or whatever) and we're on your case about it?"
He usually smartens up after that.

I don't know if any of that will help you, but maybe it will give you some ideas.