You can and many people do successfully keep their Bettas in unfiltered tanks/bowls and rely solely on regular partial/complete water changes.
However, a cycled and filtered tank does have great advantages in that you avoid the peaks in ammonia, that you may get by relying just on water changes alone.
On the other hand, a filter can cause the problems you've described and this can stress your fish.
It sounds likes you the filter is just too powerful for that small tank. A 1 gal tank, whilst Bettas can and are successfully kept in these, is still a small tank.
I think your options are to either remove the filter and rely on water changes alone or upgrade your tank to a bigger one and keep the filter you have or change your filter on your current setup.
What type of filter is it? Many people have success in smaller betta tanks by usuing an air-driven sponge filter as these are much more gentle than other types and theye can handle the bioload of one betta very well.
I would say you do have to do something though. Hope this helps a bit.