There's technically no such thing as an ABCN-accredited postdoc, as ABCN does not accredit postdocs (or any other training programs). Rather, ABCN is a specialty board of ABPP that's "responsible for administration of the examination for competence in the specialty of Clinical Neuropsychology" (per the ABCN website).
There are APPCN-member fellowships. But again, APPCN does not accredit fellowships. One does not need to complete an APPCN-member and/or APA-accredited neuropsychology fellowship to be eligible for board-certification or to be a neuropsychologist. For ABPP certification, one does need to complete a fellowship consistent with the Houston conference guidelines, including that it generally be two years in length, have a certain percentage of time devoted specifically to training in the practice of clinical neuropsychology, and include instruction/didactics in a variety of topics.
The term "neuropsychologist" is not protected in most states. Thus, in many instances, the only thing technically stopping a psychologist from calling themselves a neuropsychologist is essentially their own view of their competence. And whether or not they'd ever want to have to try to defend that determination.