Topping up with distilled is fine between water changes, but water changes should be done with whatever the tank was filled with at the start.
As you are doing a fish-in cycle, you do need to do a lot of water changes. Fish-in cycles involve testing every day and doing a water change whenever ammonia or nitrite read above zero. Using distilled for water changes lowers things like hardness which is not good for the fish, which us why they should be done with the same water as when the tank was first filled to keep the levels constant.
As you are doing a fish-in cycle, you do need to do a lot of water changes. Fish-in cycles involve testing every day and doing a water change whenever ammonia or nitrite read above zero. Using distilled for water changes lowers things like hardness which is not good for the fish, which us why they should be done with the same water as when the tank was first filled to keep the levels constant.