For my 5 gallons, even, I do weekly 100% water changes. In a 3 gallon, a filter is pretty much over-kill. Bettas do not need extra aeration if they get weekly changes, and too much bubbles/current will definitely stress most Bettas in a small tank.

I raised *80* Betta juvies in 2 gallons each, with water changes at least once every week, usually twice every 10 days. They had no extra aeration, and although they very rarely didn't eat every bit of their food twice daily, there would sometimes be food left at the tank bottom that I didn't bother to clean out. I changed each tank 100% about every 5 days with aged & dechlorinated water, and all of them grew up big & strong with lots of HM's (a large tailspan that is very closely related to the quality of water as they grow) and not a single case of finrot or disease. Because of that, I'm pretty confident in my water changing style =0) and I see no need in syphoning out the bottom and such in a less than 10 gallon tank, when you can just scoop him out in a cup, dump out the dirty water, scrub the tank a bit and refill with clean water, and you're set for the next week! It is not as messy or difficult and IMO the water looks much better with regular 100% WC's. A pinch of aquarium salt & a square inch of Indian Almond Leaf healed up any minor tears within one week.

Just for the record, it takes an excessive amount of time, effort, and supplies to get beneficial bacteria going in a small tank. The crap that sits in the gravel is not beneficial unless the tank has fully cycled, meaning the ammonia is 0, the nitrates are 0, and the nitrites are thriving which almost always requires a filter. Otherwise, you pretty much just have a bunch of crap in the gravel - not beneficial bacteria.