It sounds like your step daughter is unable to know how to get her needs met and maybe doesn't even know what her needs are so is behaving in ways that provide her with some sense of control. If her home life has been unstable, particularly recently, maybe acting out in negative ways reflects how she feels on the inside and lying about her actions provides her with a temporary, though not successful, means of controling the negative responses from others. This is just my two cents, but when she is responded too with scolding, maybe that validates any negative thoughts she has about herself so reinforces what ever internal struggles she is experiencing. I think people of all ages are most comfortable with the familiar, even if the familiar isn't positive. So in a way, maybe this is the need she is trying to have met.
Counseling is a good idea. Have you and her biological parents talked about why she might be lying? There are some protective factors for youth that promote resiliency and decrease risk factors for negative experiences/actions. If she has caring and supportive relationships, high (but realistic) expectations, and opportunities to participate, and if these 3 protective factors are a daily goal for all caregivers, she might start to turn around over time. It sounds like she has some insecurities so maybe these steps, along with counseling, will help boost her confidence so she won't feel the need to act out and instead act towards something that makes her happy and feel worthwhile.
I have lots of experience working with troubled youth and have come accross my fair share of pathological liars. And I can say with absolute certainty that once they were able to identify on heir own but with caring support what was troubling them, they were able to make sense out of their internal experience and eventually thrive.
It is frustrating to be in your position for sure. I might advocate for all parents to be on the same page and to help eachother take breaks when it becomes overwhelming. What this little girl needs most is discipline, not punishment, consistency, and the knowledge that all boundaries you are setting with her is for her well-being. Who knows, maybe she even has some ideas about what might help her stop feeling the need to lie.
I hope this helps some:)