Twice weekly 100% water changes with sterilization of tank and all contained objects other than the fish?
Filters themselves don't cause cycling, they're devices made to force water through a mechanical and chemical filtration to remove particles and molecules that can be caught in the fine pores of activated charcoal (kilned bone shattered into flakes, its still a mechanical filter just for actual molecules larger than water)
If you continue twice weekly water changes with actual sterilization of the tank but do not sterilize the filter, the filter will begin to support bacteria, so you could actually "cycle" the filter if it has bio-media installed and you protect it from sterilization.
With a ten gallon if you keep the rocks only about an inch thick and learn how to siphon-sweep the bottom and get the filter's bio-media loaded up with the cycle bacteria you could reduce the actual break-down cleaning of the tank to once every three months and not have to actually handle/remove the fish each time you do water changes.
Since Betta breathe air you can do a 90% water change once a week using the siphon sweeper to clean alternating sides each week. All the cycle does once it is living is turn the ammonia and nitrite which are bad into nitrate which is not chemically destructive to flesh. Nitrate in high levels is still dangerous and can be harmful but compared to the 1ppm ammonia or nitrite that will kill most fish, 150ppm nitrate in normal hardness water is not immediately dangerous.
With two boys in a ten gallon tank you're looking at very little actual ammonia or nitrate. My three gallon tanks only get a water change at 30ppm nitrate and that's about every two weeks, I do 70% change out.
Betta can tolerate ammonia of 2.4ppm nitrite of 4.0ppm or so and in pH6.8 water the average veiltail will ignore 200 to 225ppm nitrate.
Don't be afraid of the filter cycling the tank, cycling is natural and not actually bad, for it to produce dangerous levels of nitrite you need to already have dangerous levels of ammonia and if you do full water changes twice a week it'll never become an issue.
Inducing the nitrogen cycle in aquariums allows for more rigorous and longer term control of water conditions and the support of much less robust fish and plants than most people on this forum keep. It also extends the time between water changes and reduces overall lifetime stress on the fish.
So it's your choice whether to use the filter to circulate the water or not, if you keep doing the twice weeklies what happens in the filter is immaterial.