Distilled water lacks mineral content that are essential for the health of your fish and tank. Distilled water has no carbonate hardness (KH), which is essential for the nitrogen cycle to oxidise ammonia into nitrate. With no KH your tank cant consume ammonia at all. You could use distilled water mixed with tap water to dilute the contaminents in the tap water and retain some of the minerals in the water. A 50/50 split would half the amount of ammonia going in the tank.
Spring water is an option as long as its not demineralised, sparkling or flavoured.
Or RO water, which you can remineralise or mix with tap water and would be a cheaper option than distilled water.
Once you are cycled, your problem goes away. You do your water change and the ammonia in the tap water quickly gets cycled out by your filtration system.