I've used a small filter in a 1 gallon set up before. But the filter got rid of the annoying protein layer that forms on the surface of still waters. That is more of an astetic problem than a tank hygiene problem. However, I did notice that I had less ammonia spikes when I used a filter, so a filter does make a 1 gallon set up a bit more "oopsie" proof when you add a filter. But it isn't vital. :)
Do you have a small heater? Unless the room you keep your fish in is at a steady 80 degrees or so, you'll need a heater to keep the tank's temp steady at around 78-82 degrees. I use the Hydor 7.5 Watt heater in my hospital tanks, but I'm not sure if Hydor still makes it. I know Zoomed still makes a 7.5 heater with a picture of a betta on it. With 1 gallon set ups, I found it much harder to heat constantly than to clean. Why won't any companies make adjustable 15W heaters? XD
As for "tiny amounts" of food, is the Aqueon pellet a standard 1mm pellet? If so, a tiny amout would be 3 pellets at a time. For my feeding schedule, I feed 3 1mm pellets twice a day. Unless you have a King or Giant betta. Then I think a "tiny" amount for them is double.
I hope this helps.