The first part of your question is a bit difficult; how can you prove that the mother lied in court? If you have those facts that show she knew the truth then (arguably) they should have been brought-up in court at that time. Court's don't necessarily want to have to re-hear cases when a party had a chance to bring the truth to light but didn't.
It sounds more like you have a change of circumstances, given your stepson's reactions at recess, but the trouble may be showing their's a connection between the anger/violence and the move and reduced time with dad. You will want to make sure dad is active and involved at school, is seeking some kind of interaction with a school counselor or, barring that, seeking outside counseling - coordinated with mother - to address the issues.
Ultimately it may come down to having a child custody evaluation done to assess the schedule that would best meet your stepson's needs.
Based on your question, I assume that a move away hearing has already been conducted. If this is incorrect, you should speak with qualified counsel about filing a motion and seeking a move away hearing.
If the move away has been granted, then you probably looking at a "change of circumstances" standard. Collecting the evidence that you have is a good start. Stay involved in the child's life. Continue to reach out to school counselor's, teachers, administrators, etc. Has the child been a part of an IEP(Individual Education Plan)? If so make sure that you are in contact with the team members and you and your husband have good communication with them. Document what is going on and try to have some good educated opinions as to how to fix it.
Sit down with an attorney that you trust and go through the evidence to determine where you are and where you need to go from here. Remember that the child is going through a lot and needs adults to make good decisions that well help. Good Luck!
The communication above is provided for informational purposes only. It is not legal advise and does not create an attorney client relationship with anyone. Please consider the information above and get competent legal advise from your trusted advisor.