I don't think you will find that it is the pure water doing any damage to your tee shirts! In over 4 years I've never had that problem and I wear the same uniform polo shirts all the time.
Pure water isn't THAT pure, it's only marginally purer than rain, we measure it's purity in parts per million for window cleaning purposes, but it still has contaminants in it.
Put some in a glass, stick it in the freezer and it will freeze at just below freezing, have some water that has a reading of zero parts per BILLION and you will have to get the temperature down to several degrees below freezing before it will turn to ice.
Pure water will not effect paintwork or rot wood any faster than any other kind of water, it isn't on there long enough to affect the woodwork! It will dry out too fast.
Woodwork and paintwork will be more effected by condensation from dew in the mornings than from pure water (and I'll bet that the dew that forms is purer than rain too) and in any month between cleans there will have been several hours of rain for the paintwork to contend with, not just half a litre of our water once a month!!
And as for damage to paintwork, that big yellow thing up in the sky does no end of damage!
Ian
