Again, I don’t see the term ‘messianic Jew’ anywhere in Scripture.
I do, however see Paul preaching that:
“All are alive in Christ Jesus”, indicating that the Church of Acts did not regard themselves as ‘messianic Jews’, but rather as followers of Christ, and, as Paul states, members of the Church. Significantly no distinction is made between Jews and Gentiles.
If you are going to use a term from NT days, I reckon it would be “People of the Way” and later on “Christians”. Granted, there were no Messianic Jews... or Catholics then. If you are going to use modern terms, to justify calling them Catholics, I insist, Messianic Jews would be more accurate.