What you are probably seeing is your water is treated with chloramine rather than the more common chlorine treatment. Chloramine is chemically bonded chlorine and ammonia, most water conditioners will break the bond in the chloramine, releasing chlorine and ammonia. Your water conditioner will also treat the released chlorine and the nitrogen cycle quickly takes care of the ammonia.
However, in an uncycled tank the water companys use of chloramine is an added difficulty. The ammonia isnt removed through the normal processes that happen in a cycled tank.
A few points to consider.
If you are doing 50% water changes at 1ppm ammonia with water from the tap thats 0.5ppm ammonia, then the result is going to be in the region of 0.75ppm ammonia. Thats obviously still better than 1ppm. If you werent doing those 12 hourly water changes, that 1ppm in the tank could be say 2ppm after 24 hours, in which case the 50% water change would result in about 1.25ppm ammonia. So doing those water changes are still important, but obviously not as effective as they would be if your tap water was ammonia free.
As suggested, using Seachem Prime as your water conditioner will detoxify a small amount of ammonia for a day or two. This could be an important consideration when selecting which water conditioner to use while cycling a tank, and more so if your tap water is also introducing ammonia at every water change. I would consider use of Prime as your water conditioner long term as it will detoxify the ammonia from your tap at every water change.
Im going to complicate your 7 year olds science lesson a little. Your test kit for ammonia doesnt actually measure ammonia. If you look on the packaging, it actually measures total ammonia nitrogen (TAN). This is the total of free ammonia and ammonium. Its only the free ammonia side of things thats harmful to your fish. The proportion of free ammonia and ammonium that is present in the TAN you are measuring is pH and temperature dependant. The higher these are, the more of your TAN is free ammonia and therefore the TAN is more toxic. At typical aquarium temperatures you need to quite a high pH for the levels of TAN you are detecting to become toxic.
What pH are you seeing? To give you some comfort i wouldnt worry about 1ppm TAN unless your pH is in the high 8s.
As to whether the 12 hourly water changes would remove any bacterial product you are adding, then yes it probably will. Personally, as a one off, add your safestart after a water change, then dont do a water change for 24 hours and then redose safestart if you have any left. As mentioned above, these products arent the magic bullet they claim to be, but might help a little. I wouldnt sink a considerable amount of money into them. Your tank will cycle with or without them, it just "might" cycle a little quicker with them.