She's just barely five and she's trying to figure out her place in two different households. It's tough on her. She's probably looking for boundaries.
When my granddaughter she this same age, she came to live with me. We also had problems in kindergarten, although it doesn't sound like to the same extend you're having problems.
I got a daily report from the teacher when she was picked up. Basically, he would just look at her caregiver and give a thumbs up for a good day and a thumbs down for a bad day. That's all that was needed.
Each day that she got a thumbs down, she went home and spent the entire afternoon in her room. I will say that it didn't really seem to make any difference. She hated being in her room, but she just couldn't seem to keep it together at school. The entire year was rough and I don't believe my punishments made any difference at all.
The only thing that has made a difference is time and maturity. So, I think for your SD perhaps another round of kindergarten may do her good. HOwever. if she's grasping the academics, you might go ahead and let her go to first grade. I don't think she should be held back due to behaviorial issues because like I said, they will slowly resolve with maturity.
YOu just have to figure out what the consequences will be for particular behavior and then stick to it even if you don't feel like it's working. She needs consistency!