How to discourage the rude and bad behavior. Sorry kind of long

First off im going to say i love being a step mom even though it has been one of the most challenging things i have done. I have no say in medical situations(neither does my SO for now and hoping it goes well in court)or school and it is difficult to bite my tongue at times because certain things are not right nor fair but i love my step daughter.

Backstory: daughter's mom is married has one step daughter from her husband thats older than daughter and they have one daughter together that is younger. Daughter is only child in our home. Daughter is now 6 years and i have been with her dad since she was 6 months old.

My SO and i parent well together and stick together as a team even when daughter thinks she is being tricky to get her way and tries to play us against each other (doesnt happen often). Daughter is a very shy girl around people she doesnt see often or people she doesnt know. She has always been a sweet girl never causing to much trouble and definitely doesn't throw fits (we've had about 7 fits in 6 years) with her dad. The problem we've been having lately(since Christmas and her bday just passed) is that daughter likes getting presents but not saying thank you for the presents.

The actual day of her bday it was my SO's day to have her(mom had her the year before) but her mom wasnt budging and said she was going to pick up daughter right after work. Yes we are going to court for custody to have all this taken care of. Anyways to spend time with daughter on her bday my SO agreed to do something together that night. We go out for dinner and games. My SO's cousin and his gf show up with presents. Daughter opens them and doesnt say thank you. My SO of course told daughter to say thank you but she didnt. She walked to the other side of the table to her mom and immediately her mom starts coddling her telling her its ok she doesnt have to do it. Daughter also started with the crocodile tears. I couldn't believe what i had just seen. It upset me so much but now i completely understand why their children are very rude and mean.

I thank god that daughter is nothing like her sisters and is the most well behaved out of the 3. But i cant help but dred the fact that daughter is catching on to their bad behavior and starting to copy them. The older sister is very disrespectful and says alot of rude things and doesnt treat adults with respect. That is where the problem lies with daughter. Her older sister is saying she is better than her smarter than her and knows everything. So daughter believes it and is now copying her behavior such as saying rude things and not respecting the adults in the house.

How do we stop it? My SO is all about daughter listening to me and she does for the most part but every once in awhile i will have to tell her too many times to do something and i will get a smart ass remark from her (fully knowing where she got it from). I also tell my SO and he talks to her about that stuff and how she is not allowed to be rude to me because im the adult. She listens to her dad no problem mostly but lately she has been pushing boundaries knowing what her punishments are for not listening. Ive tried talking to daughter telling her its not cool to copy other people because she's great the way she is. Ive also tried telling her, when she tells me what her sister says to her, that her sister is a little girl and doesnt know much right now and what she is saying is not true. How do we stop that behavior when the adults at her moms house dont stop it. Her mom is rarely home and her step dad only pays so much attention to her (he has his favorite child that he pays attention to). Im very lost as to what to do and any advice or help would be much appreciated.