So, the vet looked the cat Astrophe over and declared that as far as she could see, his only real problem was that he was constipated. He's been given some fluids and an enema. (If I have to know that, you do too.) We're basically supposed to keep him hydrated and dose him regularly with a little bit of Miralax.
So I did make the right call, which is a profound relief.
Also? Once again, parenting is all about the poop.