In a one gallon, you don't need a filter- would create more room, the out take won't be too strong so he would feel comfortable swimming at the surface where he needs to be to breathe.
The only difference for you without a filter would be 1 extra water change a week.
With filter you would do 50% per week with siphoning of the gravel, without the filter you would be doing 1 50% and 1 100% water change per week.
So a little extra work for you vs. more room, no current (bettas tend not to like currents since it creates drag on their fins and makes them heavy), and no damaged fins.
Otherwise, what Bomba said- panty hose on intake and an aquarium sponge on out take since it sounds as if he can't tolerate the the current if he is staying at the bottom. One gallons with a filter tend to be too strong and not always idea most of the time.
Bettas are one of the very few who don't need a filter when appropriate water changes are made, and they don't need the aeration other fish need as they breathe the air we do at the surface of the water.