hi nashnut
there are many people who would say that cycling a small tank is impossible, but others who argue that cycling would be the same in a small tank as a much larger tank.
If you are planning on keeping a betta alone, many people simply do a large water change every week (around 70%) to keep the ammonia levels low, and leave the tank uncycled.
If you do decide to use a filter, for a tank that size a simple sponge filter would be adequate powered by a small air pump. If you arent in a rush and have an established tank, you can remove the sponge from the filter, and submerse it in your established tank by the gravel or near the current filter to build up a bacteria colony for your betta (leave the sponge in place for about 3 weeks) before replacing it on the original filter stand and using it in your betta tank.
An alternative would be to use some gravel from an established tank to give the betta tank a headstart in building up its own bacteria colony.
whatever you decide to try, cycling or not, always use your dechlorinator as mentioned in your first post.
I'm not sure about using bottled water having never done it myself.