I think using it as-is would probably be a waste of time, as already mentioned. One big question that comes to mind is how you'd get it in the water cooler without lots of splashing (and therefore oxidation).
If you were willing to modify it to recirculate the wort into the top of the water cooler you might be able to get something usable but without some serious research/modifications to ensure the cooling element is constantly active and able to sufficiently cool the circulating wort, I highly doubt you'd be able to chill the wort at a reasonable pace.
I imagine at best you'll probably spend about as much time as an in-sink ice bath just with additional risk for oxidation and/or infection.
All of that said, if your goal is to minimize excess water usage at all costs (and the water cooler is fully at your disposal), modifying the water cooler or at least repurposing the cooling element could at least be worth some further research and, if that goes well, a solid attempt (and answer here for the greater good).