Update: I took Tyson in yesterday. I got a week's worth of Baytril to try. She said I should start seeing results within 48 hours. If they do work, then I'll need to get a couple more weeks worth. He gets a tablet every 12 hours, so he had one yesterday evening and one this morning so far.
They were not able to do the blood test because Tyson needs to be fasted for 8 - 12 hours first, and he had eaten yesterday morning (I was unaware of this and the previous vet didn't mention it in my last visit).
So the plan right now is if he doesn't improve by Saturday, I'll bring him in for the blood test. This morning he did have pretty normal stool, but I don't want to get my hopes up in case it's a fluke.