How long does it take to refresh the entire water supply in the Earth's atmosphere?
The term you are looking for is called residence time or residency time.  This UCAR Science article suggests it is about 9 days on average.  This should be verifiable by dividing the average global precipitable water by the average global daily precipitation.

I don't see any reason why the residency time of water would offer any hint of weather predictability limits. If the equations to predict evapotranspiration used in forecast models have skill, shouldn't the 9 days not be the limit (and how limiting a factor it is would depend upon how skillful those equations are)? The residence times of vehicles on roadways is probably near/less than an hour, yet we can forecast traffic quite reasonably well in advanced in at least many circumstances (and that's prone to the much more volatile nature of human choice/error)!

Indeed I think the "residence time" of energy in the lower troposphere, if such an idea even has sense, may well be shorter than a day for the Earth given the continual heavy IR absorption and reemission rates in the atmosphere, and the dominant input being solar energy. But we can forecast much longer periods than that well.  We have great understanding of insolation into the Earth-air system, and then quite solid skill at predicting its transfer, even with the complexity of clouds and other feedbacks.  So we can outdo such a timescale even if energy passes through the system indeed quite quickly.  Likewise, there seems to be no reason for the 9 days to be a particularly meaningful limit on the atmosphere.