Well, I would think of it from your Betta's viewpoint...would he or she like filtered water?
There are some experienxed people that do run a no filter tank for a betta. They either know a whole lot about aquariums, or they have been told that Betta's come from rice paddies in nature so they don't need a filter, which is what the really knowledgeable people know how to create with the right balance of elements.
I would never try to go without a filter.
I'm too lazy to do the many many more water changes that would be required since I do not know how to run a no filter tank.
Does the tank have a lid?
Does it have a special place around the top rim where a filter used to be? I had success with a Tetra Whisper filter fitting into the opening of a tank with a lid. There are other brands that offer hang on filters if you are not working with a lid.
Finally, about using an air stone and foam filter with air line. I think, am not positive, that these can cause quite a bit of water agitation. Betta's, coming from a natural environment that does not bubble, need calmer water conditions. You will probably want to look into a more gentle filter. (Sometimes you have to get or make a baffle because even the small filters can move water swiftly). You can ask for help picking out a filter, I'm sure someone here on the forum can help you choose. Shouldn't be costly compared to pump, tube, and foam combo of a bubbler. Finally, if you need a baffle and choose to buy one, Fluval has a replacement part for their Edge aquarium that I use.
Sorry if I got carried away; I am quite fond of Betta's. It seems like a lot to go through, but....They can live a long time with the right care and conditions.