Michael Bird cites William Campbell as follows:
“Because Paul cannot yield on this point [the gospel is available to Gentiles without having to proselytize] does not mean that he opposed all things Jewish or that he would discourage Jewish Christians from following a Jewish lifestyle after they had become Christians.”
I would say it depends on the context, time of his career, and what Paul was trying to accomplish. He was never opposed to “all things Jewish” per se, but salvifically speaking those things were empty. In the ancient Mediterranean, a convert’s heritage was always worthless in comparion to the present (“patronal synkrisis”, as Zeba Crook calls it), and in cases like Paul — where the patron deity didn’t change — the stakes went even higher. In these cases the convert’s heritage was excellent on its own right (Philip 3:4b-6), but worse than worthless in comparison to the present (3:7-11). That’s why Paul says that “all things Jewish” (as Campbell puts it) are “shit” (3:8). In the honor-shame milieu there is no second-best, even if Paul eventually discovered — as conflicted converts often do — reasons to celebrate his original allegiance (Rom 9:4-5; 11:1-32).
Understanding the duality behind patronal synkrisis, I think, is part of the key to understanding Paul’s ongoing tensions with the law and “Jewish way of life”. He defends it in some places (it was, after all, given by his God); he attacks it in others (the spirit accomplishes what the law tried to do but failed). He lives like a Jew when it suits his purposes (nothing wrong that in itself), but just as easily like a Gentile (since the law is obsolete).
On top of this, he wasn’t always consistent about the matter. The Paul who insisted that Gentiles should forsake their Christly freedoms when in the presence of unbelieving Jews, and abide by minimal Torah standards (Rom 14:1-15:6, spotted clearly by Mark Nanos) is different from the Paul of Galatians. In Romans he wanted to maintain ethnic distinctions, because he realized that his former strategy — eliminating distinctions between Jew and Gentile (Gal 3:27-28) — was doomed to fail. Philip Esler explains that attempts to erase ethnicity simply encouraged groups to push for their ethnic patterns even more, and aggressively. So by the time of Romans, for a variety of reasons, Paul was playing more fair ball with Israel and the “Jewish lifestyle”.
Mark Goodacre has his own comments about the Campbell passage, and he concludes that
Paul was not ‘teaching all the Jews who are among the Gentiles to forsake Moses, telling them not to circumcise their children nor to walk according to the customs,’ but the charge had a ring of truth to it if one observed Paul’s actual practice.
… and if one listened to his inflammatory rhetoric, the degree of which depends on the church he’s writing to.