Update- Nice Surprise!
He's doing better! So much better that I decided to give him a reprieve from fish jail and let him out into the tank. I got home last night and found that he's no longer resting on his side like in the previous pics, but on his tail (more upright!). He's swimming better, not at all struggling to get to the surface, and starting to eat even more.
Still no name.
Thanks again for all the kind words. It really is amazing what fresh water, some TLC, and a dedicated community of betta keepers can do.