By way of an update, I have been using the cooler for about 4 months now. No sign of any corrosion, but the main problem is how feeble the cooling is.
The physics of evaporative cooling is too complicated to waste a lot of time with, but the bottom line is that the best my cooler will achieve is 5 degrees below the ambient temperature, when the humidity of the surrounding air is very low.
And because air is light and the battery is heavy, it won't cool the battery if it gets too hot. Essentially all it can do is slow the rate at which the battery warms up on a hot day.
I have put a couple of plywood panels either side of the battery to improve the air flow. I turn the cooler on first thing in the morning, when the ambient temperatures is 20º or below, and I leave it on until about 9 pm. The water hopper only contains enough to last 4 hours, so I can't leave it on over night, and when I am away during the day, it is again unusable.
Bottom line, if I am home on a 38º day I can run the cooler and keep the battery below 35º until it is fully charged, but on a 40º day my over temperature protection will kick in and the battery will stop charging at that point and for the rest of the day.
For the number of times that happens, I am not sure it is worth spending a fortune on cooling just to achieve a few extra amp hours. Rather I am more inclined to buy more second hand solar panels to charge the battery faster while it is still cool in the morning.