IOTL there were Jewish groups that recognised Jesus as the Messiah, or at least as a very important Rabbi: these continued to exist until quite late, probably well into the sixth century. It's been a while since I read up on these, so I can't give you any names for the groups: but they did definitely exist and flourish in "early Late Antiquity".
Leaving aside the fact that "mainstream Judaism" wasn't really a thing in the first and second centuries (a lot of what we recognise as Jewish doctrine now is a late antique reaction to the rise of Christianity), were the Jews really so opposed to proselytising? I have my doubts about this: why else would the Torah have been translated into Greek and Aramaic amongst others in the Hellenistic period if this were so? Why would there be large Jewish communities in Alexandria and elsewhere if this were an insular community?
It's better to see Christianity and Judaism as gradually mutually moving off in different directions in a slow process between the first and fourth centuries, I reckon. There were certainly Christian sects active well after Constantine that were very "Jewish" in outlook, and Jewish sects that venerated Jesus as at least a great teacher, if not more.