I have always read Gen 3:16 as, “I will greatly increase (multiply, etc) your pains…” which would seem to imply that labour pains were there beforehand, but now God’s sort of ramping it up, as it were. But I don’t know enough about Hebrew to take it back to the original language, so that could be a translation error. But I would think that, if any pain was the curse, Scripture wouldn’t use the language we find in Gen 3:16.
Another interpretation that I’ve heard is that, in Rev, Mary’s pain and wailing is not physical, but emotional. She weeps for her child (if you had to give birth in front of a dragon, you’d be scared, too) and for the Church, and “the woman” as the Church weeps.
That’s just me and my pious speculation, though. I would like to think Mary had a swift and easy time bringing Jesus into the world, but perhaps she wept to know that she had no better place for the Son of God than a stable. She wept because she feared He would be cold, she wept because she was frightened of what people would think. There are many pains Mary could have felt that were not physical. But I think, this side of Judgment, we won’t know for sure.