I think that most of the time it's not effective for one parent to dictate the consequences at the other house. Her deciding to ground her daughter and then wanting her ex to enforce it impacts your home. Similarly, if he made a choice that he wanted carried into her home, she would probably resent being dictated to, especially if it hindered her plans.
I think that while her father can back up her mother, her mother may need to accept that his consequence for their daughter's behavior might need to be different than hers. Maybe instead of being grounded during the short time she sees her dad, the child could do yard work for x time per bad note sent home from school. Or sit out part of an event and have to watch the other kids get to play first because she "owes" her dad x time for each bad note.
I also think that the father, if he isn't already, needs to get involved directly with the teacher and discuss his concerns and have her discuss concerns directly with him. Most teachers these days have email.
Each kid is different. My stepson plays video games so sometimes DH would take the cables for his game systems and let him stare at the paperweights they became. My stepdaughter is social, so sometimes we would tell her no party. Find your stepdaughter's currency within your home and go from there.
I don't remember a time when we asked BM to enforce a consequence in her home. I recall incidents she was told about, but what she did or didn't do was her decision. It's hard when there are two homes and two sets of "rules" but kids deal with different environments all the time (like school vs home vs grandma's house). If you wouldn't take a kid to grandma's and expect grandma to ground her, then why do that to her at her father's house?
Also, if the behavior is limited to school, then try to get to the bottom of it. Is it always with the same kid(s)? Same situations? My stepdaughter used to get bored and read non-school books during math class and other inappropriate times, or try to show off during a quiet time, etc. We worked with her on ways to help her remember how to behave.
I hope that made sense.
I agree that Dad and Mom should talk directly, but there's no reason IMO, that you can't give Dad input first.