I would stay with the tap water. A level of 0.5 ammonia is not that high, and the live plants and bacteria will easily deal with this and very quickly.
The initial influx ammonia at each water change is the only real concern, and to deal with this you might look at using a water conditioner that detoxifies ammonia along with chlorine, etc. Some do, it will say on the label. They do this by changing the toxic ammonia into non-toxic ammonium. Plants grab ammonium fast as their nitrogen source, and bacteria will take up ammonia/ammonium interchangeably, which ever is present.
Some might correctly say that even this latter suggestion is not really necessary, given the low level, but it may be best to be cautious rather than risky. Ammonia is highly toxic to fish. In water with an acidic pH, ammonia automatically changes into ammonium, but you have slightly basic (pH 7.4) water so I would use the conditioner and feel safer.
Another problem with bottled water is that it may contain other stuff as bad or worse. "Spring water" is likely to have minerals in it, and these add TDS (total dissolved solids) which certainly affect fish. "Pure" bottled water such as RO or distilled can work but over time it will deplete the aquarium of necessary minerals. Your tap water is probably better suited all round.
Byron.