I've seen some tiny tanks with tiny HOB
filters at the fish store. The flow was low but so was the bio load.
They were completely cycled and looked great. You could do the same thing. Get some gravel and cycle it carefully. Water changes would be a breeze.
The hardest part would be treating the water you add. You'd need a graduated syringe that measures chemical in cc's.
My understanding is Betta's do best at temperatures around 74-78 degrees f (23-25 c) If your house stays that warm in the winter you're okay.
Otherwise, a heater for that small a tank/bowl would be tough.
I say go for it.