I just checked the NYPL catalog and found The Turner Diaries, Camp of the Saints and Mein Kampf to be available. Just three examples, but I find it hard to imagine what else they’d refuse to carry if those are fine.
Is there a particular book you have in mind that “Park Slope progressives” would find too objectionable for libraries to loan out?
NYPL isn't BPL for what it's worth. :) NYPL only covers Manhattan, Staten Island and The Bronx. Brooklyn has Brooklyn Public Library and Queens has Queens Public Library. All separate systems that existed before NYC was united in 1898 and stayed separate afterwards.
Can you check if they have The Transgender-Industrial Complex[1]? It's one of the few books Amazon refuses to sell. You can only get it from the publisher or Barnes and Nobles.
Is there a particular book you have in mind that “Park Slope progressives” would find too objectionable for libraries to loan out?