How an extremely competitive, but non-accredited "regionally" program will be perceived in a phd applicant application?
I would simply mention that you were accepted to a highly selective program, providing the name.  Hopefully, the institution is well-enough known that you need not go any further.  As for the Distance Education Accreditation Commission, it is sufficiently recognized by the major US agencies (e.g., US Department of Ed and Council for Higher Education Accreditation so I think you are sufficiently covered.