A few points:
1. There are a lot of songbirds by Park Place. I've recorded 59 species on my block on Park Place.
2. Outdoor cats are a huge cause of bird mortality, including of legally-protected neotropical migrants: http://www.nature.com/ncomms/journal/v4/n1/full/ncomms2380.html This article is from before that analysis: http://www.nytimes.com/2011/03/21/science/21birds.html
3. The problem is not just outdoor cats on our block, but more generally. Jamaica Bay Wildlife Refuge, Plum Beach, and other places in Brooklyn used by migrating birds as vital stop-overs, have large populations of feral cats. I have personally observed cats there, and in Prospect Park, killing migrant songbirds.
4. Come on, I didn't say anything about "euthanizing" anything, and I'm not a vegetarian, and obviously I don't think hawks should be. Rather: if you introduce a skilled predator into an environment, you're killing birds, just as introducing snake-head fish into the pond in Central Park (as someone has apparently done) kills fish. When you release cats for "TNR," you're killing birds. And that's the case even if you're trapping cats that are outdoors. If someone collects land-mines from a road, and then puts them back where they were, and people die, they've caused those deaths.
Please consider not doing it.