Floating plants are advisable too, because they are fast growing and that means they will take up a lot of ammonia/ammonium.
There is something here we are not seeing. I also thought about bioload, but those four fish are on the small side, and the tank is large enough to easily handle the bioload and more. The water changes have been frequent if the info in this thread is correct. The tap water tests zero for ammonia, or so it was reported earlier (?).
Do they add chloramine to your tap water? This is a bonding of chlorine and ammonia, and conditioners break it down but the ammonia is there, though usually it registers around 0.25. Bacteria and plants can easily handle this. But the fact remains, there is something causing ammonia, unless the tests are not being taken correctly. Be very careful when adding drops to the test tube, just one drop more than needed can distort things.