On my cycled tanks that are 6g and less, I make twice weekly 50% water change and only vacuum with one of the twice weekly water changes, I only swish the filter media in the old tank water when the water flow has slowed to keep the good bacteria alive and to prevent a mini-cycle, I don't change the filter media unless it is falling apart and that is 1-2 times a year.
Remember to use a good dechlorinator with all new water added if you are on city water supply and keep the water temp within a couple of degree with new and old water so not to shock the fish with sudden temp changes.
Regardless of filtration you still need to make regular water changes to keep water quality at its best to prevent health/fin problems.
Freshwater fish need fresh water to thrive not just survive and to live a long and healthy life.
Sometimes bettas will bite themselves because they hurt and ammonia can be painful.
You know your tanks are cycled when your water prams read: ammonia 0ppm, nitrite 0ppm, nitrate 5-10ppm, for several days without a water change.
IMO/E-anytime you test your water and you have reading of ammonia or nitrite 0.25ppm or greater make a water change until they read 0ppm and if you have nitrate 30ppm or greater make water change with good substrate vacuum to keep at 5-10ppm.