OK, this clears some things up. First, on the ammonia...this is most likely due to the chloramine. And since this tank is thick with plants and some fast growing among them, don't worry over the minimal ammonia. The plants will easily handle this. Chloramine is a chlorine/ammonia bonded substance, and the conditioner breaks the bond but the ammonia remains, it is not removed by conditioners. This is the job of the plants and nitrifying bacteria, and they should bee able to handle this. Do an ammonia test a day after the water change, and a day after that, and see if it registers. i suspect it won't. Problem solved.
Second issue, the neon deaths...I don't know. This fish is very frequently in rough shape these days. Provided the cories remain apparently unaffected, I would just monitor things and see where it goes. It might be advisable to get a different fish if the neons do not do well.
On the Prime, I would not myself use this conditioner. It has chemicals in it that do things that really are best left alone. I won't get into all that, and not everyone will agree, but facts are facts. The API Tap Water Conditioner is safer and all you need here. I am not saying the Prime is the issue behind the neon deaths, but I will absolutely say it may be a debilitating influence. Characins are especially sensitive to all chemicals. Pure water with as few "additives" is better.