I use a filter in 5 gal and up. Alhtough I have two 5 gals that aren't filtered, I like to use a filter only if there is more than one betta in the 5 gal. So my 5 gallon has 3 bettas in it (they're separated) so I usually put a little filter in there, a sponge filter and I set the bubble to really low using the control valve so there isn't much current.
Also, on my 5 gal, I usually go 2 weeks for a water change and they're fine and my levels are fine. So, perhaps you're changing the guy's water too much? But to telll you the truth, bettas are like animals in the way that they can be one of those animals that just gets sick all the time. I have one like that. His tail never seems to heal and it seems like we're always fighting off fin rot.
Well, it's your choice, but with one betta in a 5...I'd say no filter and lay off the WCs for an extra week (but don't hesitate to add new water when the levels go down!)
Also, have you been adding aquarium salt to your tank water? This can really help the betta fight off disease.