I will give you advice that my fish health professor gave us in class
Bettas (or any fish), are poikilothermic (cold-blooded, which you prob know). When placed in water that is too cold OR too warm that suppresses their immune system due to stress because they don't regulate their temperature like you and I. They can live at those colder temperatures but it doesn't mean their body performs best at them. If you can control the temperature to around 78 or so then that is the optimal temperature range for a betta. Basically this means that is the temperature their body will best function at, reducing chance of illness. Environment is the key! If you can control this you have a better chance at biting illnesses in the butt!
Now if you find your betta is already ill, keeping the temperature warm and stable will help his immune system fight off the illness instead of trying to function at temperatures his body is uncomfortable with. Basically you are allowing his body to work less hard.
I hope this helps, if you choose not to fix the temperature, just be careful to at least keep your temperature at least stable. Fluctuations of more then one degree a day are extra stressful.
Sorry my message is so long! LOL
Your water changes seem good so I don't see any other water problems that could contribute.