Hi Doomlord, welcome to the forum.
First, never believe anything they tell you in the pet stores. Honestly. They are, for the most part, just employees who work with fish, not genuine fish keepers. In a small tank like your 1 gallon, I would recommend changing the water every other day or, if this is more feasible, 1 50% and 1 100% a week.
Just because the water is clean, like in your sister's tank, doesn't necessarily mean it's healthy for the betta. Fish excrete ammonia as part of their waste process. The problem is ammonia is highly toxic to them so it's important for us, their owners, to change the water often to remove the ammonia. We can't see it but it's there. I'm glad your sister's fish is doing well, though. Must be a hardy fellow. :)
I also know heating 1 gallons can be hard but if it is at all possible, you will probably want to find a heater. If you can, upgrading to even a 2 gallon tank so you can heat it would be good. If you can't, and I understand that sometimes it's just not possible, try to keep Doomlord in the warmest part of your room/house. If he is kept warm, his immune system will be stronger and he will stay healthier.
You mentioned Doomlord has a bit of a belly? It sounds like he might be bloated, which is a little concerning to me. Don't feed him for a day, no matter how much he begs. Also, is he pooping? I know it's a gross question but if he isn't, that might explain why he is looking a little fat.
As for food . . . hmm. How does he act when you feed him pellets? Does he spit them out several times like he is trying to eat them but can't? Or does he flat out ignore them?