It's not uncommon for fish to fight ich naturally if given the right conditions. They require a balanced diet with heavy feeding, often supplementing the food with products such as Seachem vitality will help their immune system fight it off.
With that being said - ich will remain in the aquarium as long as a fish is in there, whether they appear to have symptoms or not. The only way to completely remove it is by going several months without fish in there, and the existing fish would need to be treated in a separate tank as you are correct the copper will kill the inverts, and prevent you from ever keeping any again.
Either option is acceptable and neither is right nor wrong, I use the first option and have for many years, it is not a guarantee, but neither is treatment.