The New Perspective, at the basis (or as it seems to me), is taking Paul out of Christianity, and placing him into Judaism. Now, the argument isn't that Paul was a good Jew, or anything like that. It simply is taking Paul and putting him into his Jewish context.
But as Dunn, who coined the term according you your link (thanks for that, by the way; I enjoyed reading it), points out, there is a great deal of ambiguity in what things like "Jewish context" or "christianity" and other such terms mean. That is (as I'm sure you are aware) it's certainly true that we can't take our modern use of "christian" and apply it accurately to Paul, but the same kind of issues are involved with speaking of Paul as "Jewish". You may have already read it, and perhaps others active on threads concerning Paul (e.g., AE, Jayhawker, etc.) have as well, but I've uploaded Dunn's paper "Who did Paul think he was? A study of Jewish-Christian identity" (NTS vol. 45, 1999). For those interested the paper is quite related to the OP's topic and is a relatively short but nonetheless nuanced approach to the question of Paul's socio-cultural and religious identity (and I apologize about the site I used to upload, but I don't know much about file sharing and when I joined this forum and needed to share sources I just went with the first free site that came up with a google search).