Well, I went out and tried to gently pry it away with a long stick. It just kept shoving its little talons back in the screen. I didn't really want to touch it but after Animal Control pretty much said "deal with it yourself" I kind of thought I didn't have a choice. I got the ladder out and set it up by the window, and went inside for a towel.
But wait! NF friend to the rescue. At that moment, she called, and I happened to mention to her that I was screwing up the nerve to go grab a live bird off my kitchen window screen. She said "Wait, I'll be there in five minutes." So she came and went up to help the bird, but it flew away before she could touch it. Weird.
It flew to the bottom of a nearby tree and it was just sitting there, not flying away when we went to investigate it. It didn't look injured-- just a little dumb or dazed or something. We were contemplating taking it to the wildlife refuge when it took off into the woods.