I rarely enter a Betta thread, but the title caught my attention and I would like to offer some comments on the issue of filters.
From the photo of your Betta (a beauty) and tank, I would leave well enough alone. There is no need for a filter in this tank. I am assuming it is heated, I see a heater in the right rear corner. I would suggest some live floating plants; the plants in the photo appear to be artificial (?) and there is nothing wrong with that, but live floating plants are quite beneficial. Your water change schedule is fine, though live floating plants and a larger (5 gallon) tank would mean only one water change per week required, and it could be less than 100% which would avoid removing the fish every time which I assume you must be doing.
I read the Betta Care Sheet and see nothing amiss there so far as filtration and temperature are concerned. I had a 10g tank running for over a year with no filter, just a heater. It had live plants and groups of pygmy cories (11) and dwarf rasbora (Boraras brigittae, 12) and received a 70% water change once a week. This was something of an experiment, and it became clear that provided the tank is biologically balanced, has live plants including lots of floating plants, and receives a substantial weekly partial water change, there is no need for a filter as far as clean water/bacteria is concerned. The bacteria colonize the substrate and even in tanks with filters theree are more bacteria in thee substrate than the filter and the substrate bacteria are the most important for a healthy biological system. The filter can keep the water clear, crystal clear; but this has no impact on the fish per say. I added a sponge filter to my 10g after the year, and it did make the water clearer. And the filter obviously is the source of water currents (beyond the normal thermal currents that are active in any healthy aquarium) which may or may not be necessary depending upon the fish species. A Betta does not need, and does not appreciate, water currents. The "playing in bubble streams" is a very different thing and not indicative that these fish like water currents.