The only thing to watch for is temperature. Room temp water poured into a tank is cold enough that any fish swiming directly through the current could get stunned and killed.
Your fish would have to be exceptionally sensitive and very weak for a little cold water to "stun and kill" them. In general, fish are actually quite adaptable to varying conditions, including temperature and pH change. If a little cold water was all that was needed to stun and/or kill a fish, the fish in the wild had better stay away from any mountain stream. Or even rain wash-off since that rain is probably not the same temperature as the stream/river/lake/pond the fish is in. I am sure that you have been outside in the summer when a quick cold rain comes up. I have never heard of a massive fish die-off after such an occurrence.
Beyond that, I know that I perform large, often >50%, waterchanges with room temperature water all the time, without any fish loss whatsoever. In fact, the cooler water often starts a spawning event for several of my fish. Not only are they not dying, they are reproducing!
Research actually shows that the most stressful change is actually a change in hardness. However, as the OP is only changing the mineral content by a tiny amount, the fish will adapt to the tiny amount pretty easily and quickly.
My only warning is that not to convert to 100% bottled water, as sometimes bottled water will be very poor in buffering capability. However, 50% bottled and 50% tap should be fine. If you really want to be sure, get yourself a KH test. So long as the KH remains significantly above zero, there will be no concerns about lack of buffering.
The minerals in the water basically have almost no impact with how much oxygen the water can hold. I cannot say exactly zero, but it is very, very close to zero. In other words, dissolved oxygen and dissolved minerals are completely independent upon on another. The amount of dissolved oxygen is much more dependent upon temperature than upon the minerals that may be in it.
