Well, I"ll tell you how I keep Bettas. Their tanks are not cycled and they don't have filters or air stones. Three of the fish each have a 2.5 G tank, which are pretty full of plants, and one has a 5G tank, with fewer plants and a very small filter, and a sponge filter. It was my QT.. now it's another Betta tank until I get my hands on another 2.5G. The plants in the 2.5 G tanks are anubias, a bit of fern, a couple of small frogbits, a lot of Hygro difformis [ water wisteria], and a lot of mosses.
I feed mostly live food, most of which survive for days until the Betta catches them, so I don't have to worry about food fouling the tank. One of them won't eat pellets anyway, so live is the only way for him, and my latest one is clearly a live food fan too.
When I test the water in the Betta's tanks, the results are always the same. Ammonia 0, Nitrite 0, Nitrates <10, and sometimes not even detectable. Because I want the mosses to grow well, I add a few drops of API Carbon Boost to each tank every day or two, and a drop or two of Flourish fert twice weekly. You could skip the carbon boost, but maybe not the ferts.
Each tank has a lamp, it's an 8 inch round reflector, clamp lamp type, with a 9 or 11 W spiral Daylight bulb in it.. 6500 K. They sit right on the tank rim or are clamped just above the rims, and there's a piece of plastic canvas on top of each tank so the fish can't jump.
I only do water changes about once every six to eight weeks, and if I used a syringe to suck up the poops I probably would not even have to do that. I top up for evaporation, using Distilled water mostly, so that minerals won't build up too much. I think I'll continue to do changes about 6-8 weeks apart because I worry about the minerals in the water being partly used up by the plants, and a change will replenish them.
In a 5G, you could have a lot more plants and rarely change water at all, just so long as the plants grow well. They will use up all the waste that a Betta produces. It's not like they are large or active fish, after all. These tanks also have a couple of small snails in them to keep glass clean. I run the lights on a timer, 12/12, and so far, no algae issues have come up, and it's been since April with the lights on 12 hours a day.
You can cycle a filter if you wish to, but if you like the look of plants, you can do it with plants alone and no filter. Obviously, for the sake of safety, you should test the water regularly when the tank is new. When I began keeping the Bettas this way, I tested daily, and the values remained very constant. Now I test maybe every two weeks.