Make sure you wash the filter out really well each time you move the fish. The filter will draw in and trap the whitespot cysts after they detach from the fish. If you don't get the cysts out of the filter then they will hatch out a few days later and reinfect the fish. Hence the reason to only have an airstone, heater & thermometer in the container.
If you are concerned about ammonia, then you can add some Ammogon or Zeolite (same stuff different name). It will absorb any ammonia from freshwater. When you clean the container just chuck the Ammogon out and replace it with some new stuff. You only need a bit of Ammogon spread over the bottom of the container.
The other thing you can do is add a dechlorinator/ water conditioner that traps/ binds ammonia and stops it affecting the fish. Ammolock is one brand but there are others. You add this to the container and it will bind to the ammonia and prevent it damaging the fish.