KH does impact fish, but significantly less than does GH or pH. The GH is the most important. Taking a step back, your neons will without question be better off (health-wise) in water with zero GH and KH and a pH below 7. As soon as you begin adding minerals for the plants, the fish will be impacted, as they will also for any increase in KH and possibly pH. Keeping this minimal is obviously better for fish. This is primarily why fish and high-tech planted tanks do not really go together very well. I remember an article in Practical Fishkeeping a couple of years back which referenced studies showing that diffused CO2 during the day (only) even had a negative impact, which frankly is no surprise but still not well accepted.
KH is primarily a buffer for the pH to prevent fluctuations. If one is adding for example calcareous minerals (calcium and magnesium) for the plants, these will raise the GH and also the pH, depending upon the amount, and then as the plants use them the GH and pH lowers. It is true that in natural waters there is a diurnal pH fluctuation, but it is only a few decimal places. If this becomes larger in the aquarium, and from my forays into researching planted tanks it seems it can, the fish will not be in the best shape. This is one reason I use Flourish Tabs for the larger swords which need calcium particularly; the nutrients are not released into the upper water column as they are when powder/liquid fertilizers are added to the water, so the fish do not have to deal with the strain/stress but the plants still get the benefit.