Some things I read online said the second night was better than the first night. Others said it was worse. So we gritted our teeth and brought the crate and the rattle can upstairs. I was super sleepy from the night before so I fell asleep almost immediately, and woke up with a start at 3am thinking "oh no, the dog hasn't made any noise and we haven't taken him out at all!" so I took him downstairs and he peed in the crocuses.
Cory woke up at 4:30 or so and took him out again but he was dry, so I think not giving him water after 5 was a good thing. Except that he was thirsty after his walk last night, poor guy.
It looks like the rattle can works pretty well - I don't like the negative reinforcement but at least we're doing it so that he can't see us. Hopefully he'll continue the quiet-in-the-crate behavior when we go out without him.