I don'tknow much about RO, but any water added to the tank should be treated with water conditioner (even bottled spring water). Used to be that you could leave the water out for 24 hours and it would cause the chlorine to evaporate but now they also treat water with chloramine, which does not evaporate so you must use water conditioner all the time. You can buy water treaters at any pet store, they make some specifically for betta fish that include what is called 'stresscoat' and helps keep the fish healthy.
Also when you switch over make sure you don't do a 100% water change, this can shock the fish and endanger it's health, never ever change more than 25% - 30% of the water at a time and make sure the new water is about the same temperature as the water in the tank. REMEMBER that betta fish must have temperatures in the 78-80 degree range all the time or they won't live very long and are prone to illness, so if the water in your tank is 78 degrees and you add a few cupfulls of 50 degree water the tank temperature will fall and the fish could get very sick.