P’Tach does not have a dorm that I know of
They also require an IEP (That I keep talking about) and a psychological report. Unfortunatly I dont think you can get these things by Septemeber. An IEP requires classroom observation as part of the report.
Unfortunatly this kind of research for a special school really requires research really done in January (or even before) rather than June.
I dont know much about dorm yeshivas, but I suspect they arent that cheap either. Dont forget they have to house and feed the students which isnt cheap and they also have to hire extra help to watch the boys at night from safety issues (Like keeping out intruders) to making sure the boys dont cause any trouble by hanging out at the movies or with girls.
IMO your best bet for the fall is probably a private tutor