You do not need to cycle a 3 gallon Betta tank if you are only keeping a single Betta in the tank, regardless of the type of filter that it has.
The answer may surprise many people with some to moderate knowledge of aquariums because it depends on a deep understanding of the biology of aquaria.
The purpose of cycling an aquarium is to establish the bacteria and other microbes required to process ammonia and ammonium through nitrite to nitrate in an appropriate filter medium. Ammonium is the most toxic, nitrite is moderately toxic and nitrate is the least toxic of the three compounds. These bacteria can colonize every surface in the aquarium but usually they are encouraged to grow in a medium that has a large surface area per volume. These are the basic facts that most experienced aquarists know. It would be more appropriate to talk about cycling the filter rather than the aquarium.
Most currently-used filters use either foam, ceramic rings or some kind of plastic media. These may be included in different forms in canister filters, internal filters or hang-on the back filters. It is the only filter material in a sponge filter. Foam also functions as a mechanical filter by removing suspended particles from the water. A small filter, such as used in a 3 gallon aquarium probably uses a small cartridge with a teaspoonful of activated carbon. It may contain a thin foam pre-filter intended for biological filtration. These filters do very little other than mechanical filtration and provide water movement. The tiny amount of activated carbon is quickly spent.
A single Betta can be comfortably maintained in a very small container, such as the small pint bowls seen in fish stores, provided proper care is given. These bowls are never cycled, and if the Bettas arrive healthy they never die before they are sold. How is this possible?
First, the fish are maintained at the proper temperature, 76 to 82 F. The fish are fed sparingly once or twice a day. Most of the employees feed a few Betta pellets to each fish but when I feed, I use a pipette to give each fish a few thawed blood worms. Once or twice a week, the fish receive a 100% water change. Ideally a few new bowls are set up with dechlorinated water at the same temperature as the are in. The contents of the bowl are dumped through a soft net and the fish is put in the new water. The old bowls will be given a quick scrub in hot water, refilled and the process repeated until all the bowls have been changed. Many bowls can be changed quickly and since the bowls are clean, they always look good for customers. Since the fish are not overfed, they produce small amounts of waste, which is removed by the water changes before it become toxic.
If you are keeping a single Betta in a small aquarium, less than 5 gallons, you can do partial water changes less frequently and achieve the same results. In any aquarium larger than 1 gallon I suggest changing 1/3 of the water weekly. Less than a gallon, change all of the water weekly and less than a quart, change all of the water once or twice a week. The relatively large water changes eliminate the need for cycling. A single Betta is a relatively small bioload provided you don’t overfeed.
This can be scaled up to any size aquarium IF you are willing to do the water changes and IF the bioload of the aquarium is relatively small. A single adult Oscar in a 55 gallon aquarium will pollute the water very quickly. Even with heavy filtration you still need frequent water changes. If you have a continual flow through of clean water you can pack the aquarium full of fish, feed them heavily and they will grow quickly.
The real purpose of cycling the aquarium is to allow us to reduce the frequency and size of water changes to something that the average person can do without an undue amount of work. Once you know how it works, you can tweak things to your advantage.
I use sponge filters in all my many aquaria. I run two sponge filters in some aquaria, each of which can support that aquarium. Say I attend a fish auction and I buy more fish than I have room for. I can instantly set up a fully-cycled 10 gallon aquarium by taking half the water from an existing 10 gallon aquarium and taking the extra sponge filter from the aquarium with two sponge filters and adding them to an empty 10 gallon aquarium. I will add 1–2 gallons of freshwater out of the tap to each aquarium each of the next 2–3 days until each is full. Instant fully-cycled aquarium! Or I can pump 7–8 gallons of water from a 55 gallon aquarium and top each up with fresh water from the tap right then. This is why I said that properly it is the filter that is cycled. What about the existing tank? I have just removed half of the biofilter capacity. Bacteria can double at least once a day, and often several times a day. Within a day, the bacteria in the remaining sponge filter have increased to handle the bioload of the aquarium, as have the bacteria in the new aquarium.
If you want to run a small aquarium as an actual aquarium, with live plants, several small fish, shrimp, snails, etc., you then do want to cycle the aquarium so you can support a heavier bioload than a single Betta. Also, bettas can breathe atmospheric air through their labyrinth organ, but for many other fish you need water circulation to maintain sufficient levels of dissolved oxygen. You may want to add a small sponge filter if the filter that came with the setup does not provide enough biological filtration. Note also that live plants will consume ammonium, nitrate and nitrate but they don’t remove the need for water changes. Plants leak a number of organic compounds back into the water which also need to be removed.