Hi backflip and welcome to the forum.
So I understand your betta has developed a white patch on his belly? Does he look fatter or bloated in that area at all? Often when they get a white patch, they bloat up too. For now, I'd like you to not feed your betta for a day. No food at all. When you start feeding him again, cut down his food to only 2 pellets twice a day. If you can, getting him some frozen bloodworms and feeding him 1-2 worms for one of his meals would be good, too. Bettas love variety.
If your betta looks bloated at all, please post back and let us know.
I believe your betta has slowed down because he needs his water changed at least once a week. For his 2.5 gallon tank, you really should change at least 50-75% of the water once a week. Ideally, it would be best if you could change it twice a week but I understand that sometimes there just isn't enough time. But if you could definitely try to change it that much once a week, I think your betta will perk up a little more.
Also, unless you live in a tropical area where the weather stays in the 80's year round, you are going to need a heater before winter. Bettas are tropical fish and need waters around 76-80 degrees so when it gets cold, your betta's tank will get too cold also. A good heater to look at is the Penn-Plax Cascade heater. Also adjustable heaters by Marineland and Marina are good but be sure they are ADJUSTABLE.
I hope this helps you. Good luck and keep us posed.