Neither of my bettas have filters. I just make sure the water is kept fresh and they are fine. The one in the split tank had the filter his side but he hated it he kept sulking up the corner.
I'm always interested when this discussion comes up. I purchased my betta from a local maidenhead aquatics where all the tanks have a fair bit of current and only a token plastic plant or imitation log. None of the bettas ever do the things I've seen described on here which is weird, this is repeated in other local LFS's where cories go mental swimming in the current from incoming water and the betta is at the other end of the tank looking perfectly happy.
I purchased my betta from the afore mentioned MA and he's in a 31L tank (9 or 10 gallons?) with the filtration that came with this which creates a fair bit of current and it has never bothered him. He goes a little pale if I switch the airation / pump on because this tends to cause much more surface disruption making it more difficult for him to get to the surface to breath. The filter runs 24/7 for the benefit of the other fish in the tank.
At the end of the day it seems to depend where you got your betta from and whats hes used to, I've seen people report bettas die from being switched to tanks with strong currents, and if we're talking about a mature fish, who has never been kept with filtration this sort of change is going to be immensely stressful, thats just logic.
You can do things to cut down the current / surface disturbance. Turn off the air pump / bubble wand if you have it - even with fish that need a high oxygen level like ottos (I have these in with my betta) if you have a filter that drops water into the tank, you may not need it. The water falls in my tank onto a piece of mopani wood which cuts the downward current into the water. Also, remember Bettas will rest in plants, on the surface etc. As long as he can reach the surface without difficulty (you'll see him stuggling) is not constantly pale and is eating, don't worry too much