Last night I figured out that my heater had been set wrong, and Mephisto's tank water was a little on the cold side. After turning the heater up a bit, he came out from his log and swam around for a little while. This morning the tank was up to 80 degrees, and he seemed much more active and interested in his new tank.
However, he's currently gone back to sleep inside his floating log, and he's been there for most of the day. He comes out every few hours to make a little lap around his tank before returning to his log to sleep.
I still haven't seen him eat anything, but his belly doesn't look distended and his fins are all flowy and relaxed, so I think he's okay. I'm hoping it's normal for a new Betta fish to hide and sleep this much, and not that there's anything wrong!