I asked my pal the local rebbe about this. The answer she gave was a typical Jewish answer (well thought out and expansive).
There’s a theory among American Jews that all Jews are converts in spirit. The urge to assimilate and leave one’s own tradition behind is so strong, that every American Jew ought to be seen as a Jew by choice, rather than by birth.
With that in mind, the Jews in this particular community encourage Jews to pursue what makes them happiest, whether that means becoming a new Jew, or leaving Judaism for something else.
The local rebbe did make it clear that she wished the “messianic Jews” would call themselves Christians, noting that it is fine to be a Christian, but you should identify with what you truly are, not what you left behind. She does not agree with being a “counter-missionary” but she does argue for “truth in advertising”.
I don’t have a dog in the Jew -vs- Catholic battles, and I’m not going to jump into the arena as a gladiator on either side. I just thought it was an interesting answer to a question that was first posited here on the RC forum.