7.6 is not high, it is just a bit over neutral but not far. I've kept fish like that in water that got up to 8.7 as the dry season hit it's peak. Do you know if you have chlorine or Chloramine in your water? If it is chloramine that would explain some of the amonia reading in the tap.
I believe there is something protective about the acidic conditions in the tank. If you were to raise the ph without dealing with the ammonia and no2 the toxicity would be much higher. It is a conundrum though, because nitrogen cycle bacteria need minerals that tend to be found in more abundance in higher than 7ph water.
Either way there needs to be slow changes in total dissolved solids in the tank water. Fish don't care about ph much as long as it is not to extreme (over 9 or under 5ph). They are much more sensitive to the osmotic pressure of water, which is dependent on how much is dissolved in the water. Change to much water when the tank a lot different than the source and you can put them into osmotic shock even if the ph doesn't change. (it often does change also but isn't the source of shock)