The idea that “Federalist Society wouldn’t recommend anyone but a hardline anti-abortionist” somehow has become common knowledge, but it is a complete fabrication. If you go to their website’s list of their events and conferences, you’ll find more about the Clean Water Act than abortion. (Indeed, after filtering for events and talks tagged “religious liberties”—I couldn’t find anything about abortion in the first ten pages of results.) Likewise for their Twitter. It’s not really one of their issues.
The idea of Federalist Society “ideological purity” is also pretty amusing, considering that there is a large Never Trump contingency within the organization: https://www.huffpost.com/entry/george-conway-starts-anti-tru.... I’d wager there are a lot more pro-choice Federalist Society members than pro-life members of the American Constitution Society.