Unless your water just plain tastes bad, by the time it is softened and neutralized, I wouldn't think the chemistry would be bad for Skeeter Pee. If it does not taste bad, I would try cutting it 50-50 with your softened and neutralized tap water. Otherwise, I would go to the store and buy bottled drinking water, not spring water.
My worry about using plain Reverse Osmosis and/or De-Ionized water is the lack of minerals to buffer the pH. The pH of Skeeter Pee is already pretty low is one of the problems that makes it a tough ferment. I know there are items you can add to help keep the pH down but I am not much of a water guru.
Worst case, the cost of Skeeter Pee is pretty low compared to some other beverages. Go for it and if what you get isn't quite right, you can play with your water more next time.
Perhaps some others can share their thoughts on water and water treatment for Skeeter Pee.