It is my understanding that charter schools are considered public schools for purposes of an IEP because they are publicly funded. If so, the obligations of the public high school you are transferring to would be the same as if he were transferring from another public school. Those obligations are to either accept or reject the previous school's evaluation and IEP. If they accept, they must implement. If they reject, they must keep the "old" IEP in place until they conduct their own evaluation.
I would suggest reaching out to the new school NOW to discuss transition and what documentation they will need. I would also ask for a meeting with the special education team to discuss your concerns regarding the transition.
Finally, since all states are a little different, I would advise reaching out to your state department of education (special education division) and ask them what the procedures are for transferring from a charter school to a public school.