Most states make a distinction between statuses that allow immigration intent (like H-4) and statuses that prohibit immigration intent (like F-1 or J-1). H-4 students are allowed, under federal law, to establish a domicile in the US, so most states have opted to treat them like US citizens when it comes to in-state residency determinations.
It seems like Pennsylvania is an outlier and does not have a binding state-wide policy on that matter. I am sorry to hear that this is giving you problems.
One word of caution if your son decides to attend the University of Pittsburgh: your son will probably age out of H-4 status before he finishes college. At that point he will probably have to switch to an F-1 student visa, which may prompt the university to re-classify him as an out-of-state student. If money is a concern, I would urge you to clarify this situation with the university well ahead of time.