What is the temperature in the bowl? Some disagree but my opinion has always been that 1 gallon is way too small for a betta. Can they live in a small bowl like that? Yes. But they're MUCH more prone to illness, shortened lifespans, etc. and tend to be less active and colorful. When you're able to, consider upgrading them to small aquariums, even a 3-gallon would be a huge improvement
Anyway, for now, I'd be interested in your water parameters. Bring a water sample (not right after you change the water, right before would be better) to your local pet store (one that sells fish) and ask them to test it. Ask about your ammonia, nitrite, nitrate, and pH. Alternatively, buy a cheap test kit and test your water at home.
Edit: What hiding places do the fish have? If they don't have anywhere to hide they're going to feel pretty stressed. That could explain Sheldon's reluctance to eat.
Also, I wanted to respond to this, you said:
"I don't understand, because Bently manages his gallon bowl just fine."
You've only had the fish for a few months so this actually isn't true. You don't know that yet
. Bettas should live a few years, a few months isn't enough time for many issues develop. Make sure you keep an eye on him.