There's little point in having a filter in such a small bowl, it's easier to simply do frequent 100% water changes.
Was this 'aeration system' you have supplied with a smallish plastic plate at the bottom? If so, it's an undergravel filter, one of the most useless, obsolete methods of filtration.
I've read your other posts about you planning to upgrade to a 5 gallon. Until then, just do 100% water changes every other day. The turbulent current from the air pump will distress your Betta, and he will do much better in still water.