A single betta in a 10 gallon, don't worry about cycle. If you add any other fish in there with him, then you will certainly have to cycle the tank because it will do it anyway if you don't.
1 betta in a 10 gallon, with lots of decorations, like live plants, lace rock, and maybe some snails for algea, is a beautiful tank. I love the idea of a betta roaming around his own space, why bother with other fish!
Certainly use a mechanical filter on a 10 gallon, just use a small filter and set it on low. The little bit of water movement will help keep the water cleaner and him healthier than sitting water. Any small HOB filter would do, and Mr. Betta will have enough space to avoid the filter current when he wants too.
See, the problem with POWER filtering most smaller sized betta tanks, like a 2 or 3 gallon tank, is that there is not enough room for the betta to escape the current, so he has to constantly be swimming. In a 10 gallon tank, he can avoid the filter current and make a bubble nest.
Now, I use jars for all my bettas for now to avoid spreading sickness amoung my adult breeders, plus I have so many young adults and fry I need to have jars anyway, so its not like I would be saving any energy. But even my jars are large, with java moss and regular water changes and attention, so its not a terrible life!
I am planning a custom-made, heated, shelving unit system, for my betta collection. about 10" wide by 12" deep by 5" tall (2.6gal) trays, that slide-out like dresser drawers in a 6' tall by 48" wide cabinet. Each row will have lighting, and the cabinet will have a copper-tubed radiant heating system so I can put it in the basement. Its probably goind to cost me about $600 to make, so I am just in the planning stages for now.