If the father of a child was granted full physical custody (but both mother and father have joint legal), and they live approximately 45-60 minutes away from each other, and the mother files for joint physical to request every other week with the request that she would drive the child to and from school every day on her weeks, what is the likelihood that a judge would agree to this?
The mother lost custody a couple years prior due to withholding visitation from the father and filing false CPS claims against him for abuse, and for expressing severe antagonism towards father and stepmother that was proven in court. The guardian ad litem also stated that the child was better off with the father and so he was granted primary custody with visitation to the mother. State: Virginia. The child is 7.

Unless she can prove to a court that she has changed I doubt it the court will change custody. Also, joint physical custody (every other week) will not work unless both parents live in the same school district. Even if mom drives the child, a one hour commute is not going to make the court happy.

The mother is planning on moving into a house in March, and that will be the only "change" that she has made since custody was changed over to the father, and she seems to think that if she agrees to drive him back forth, the courts will allow her to have joint week on week off.

I think she's wrong. She seems to have lost custody over behavioral issues. The court would need to be convinced they are no longer an issue.