i've kept bettas for 15 years and i've never kept them in anything bigger than a 1.5 gallon bowl with no filtration and no heater, and i do a 100% water change once a week and i've had bettas live for 4-5 years, which is an average life span for a betta
That's what my mom does, it seems to work for her.
For a filter, maybe an Aquaclear Mini?