Op, you DD isn’t diagnosed with any condition yet. They are displaying some signs of potential conditions, but also that she is making some very poor choices. The sanctions will be for poor choices rather than just forgetting her pens etc.
I’m not sure why a school would want to make it a managed move because, if your DD is as disruptive as you’ve explained, they’d probably be glad she was no longer their so the other students weren’t disturbed in their lessons.
School rules are there for a reason and parents need to support the schools in enforcing these, regardless of whether they agree with them or not. To undermine the school and its sanctions, especially in front of their children, prevents the school from operating in the best possible way for all the students that attend.
The average secondary school has 1000 students - what would happen if there were no rules? Additionally, the rules are there to provide learning as to what will happen when the children leave education, and the comfort of home, and enter the real world.
I wish your DD all the best at her new school, but please do not believe that all will be fixed ‘if’ your DD is diagnosed with ADHD. Poor decision making is still poor decision making and will results in the same outcomes.
You’ll need to support her with this as a parent, along with supporting the new school with all of their rules (parent-school agreement which you agree to when your DD starts at the school).