Is there a possibility that overturning Roe could cause pro-choicers to give up in the several years subsequent?
No. I don't think there's a single developed country where abortion policy is not met with opposition one way or another. (I mean, I could be completely wrong, but even staunchly Catholic, anti-abortion Poland has a respectable pro-choice group numbers-wise.) This isn't like gay marriage.
Even Ireland, for that matter. Does Denmark have a strong anti-abortion community?
No. In Scandinavian countries the issue is considered settled, since the pro-choice share is 90%+. Christian Democrats and church groups focus on reducing the number of actual abortions by improving conditions for young mothers and giving better information about support opportunities. Norway has the strongest anti-abortion sentiment.