To get a change in custody, you are going to need to get a court order. Based on the nature of your situation (your child's mother is planning on relocating, and your current custody arrangement has your daughter changing at the end of summer - which would coincide with a change in state/change of school at that time), I would highly recommend hiring one of those attorneys you spoke with.
Your daughter is old enough that her interests/wishes are going to be taken into consideration (CA gives consideration to a child's interests in a custody placement when making a placement decision in the best interests of the child - although this is not the sole determining factor).
The other factors you identified will also be taken into consideration (child is not doing as well as she once was, she is facing a potential change of state/change of school, etc.).
But you need to move quickly in order to deal with this prior to the end of summer.
Similarly the child's mother needs to deal with these matters as she likely needs to be filing a change of state application.
The ideal resolution would be for both of you to reach a resolution through negotiations or mediation, but you need a lawyer on board now to be able to put things in place to have a court order ready by August if this doesn't happen.