As long as you are getting good water with no nitrates, ammonia or nitrites, I'd do a partial ( or major, depending on whether the nitrate level is above 20PPM) water change just to reduce the level to as close to 0 as possible for starters.
What I believe was happening is that since the tank was small and the "bioload" was small ( 1 small fish), the biological bed is very small so any additional ammonia/ ammonium from your source water just causes the bed to grow until it outgrows the amount of ammonia/ammonium. HOWEVER, it will always die back when the ammonia/ammonium level reduces which is why it's not always easy to keep small Betta only tanks stable. So to reduce the fluctuating ammonia/ammonium levels in the tank, you need to use water that doesn't have any of it. As I said, Spring water is the best choice for this.
Also keep in mind that since the amount of waste being produced by such a small fish is low, the water quality will take time to decay in a 6.5 gal tank so there really isn't a need for massive water changes unless you are over feeding. This is a very different situation than your 29 gal with many fish in it. This is why you need to treat the tanks differently.
Some low light level plants can also help keep the nitrate level low as well. If you like Pothos, a lot of people keep them near their tanks so that the roots are in the aquarium and absorbing the nitrates. Just another way to help reduce your need to change large volumes of water. Water changes are still necessary but the volume of the change can be lower at each change.