Sounds like a hassle.
I am not terribly familiar with the coolers that chill bottled water, but some years back when I was a lad I got my hands on a pair of water fountains from the old school. They were designed to chill water for drinking. The amount of water needed to chill to slake a thirst is MUCH less then the amount of water needed to chill a batch of beer.
If that cooler is anything like that water fountain, there is no way it could keep up with chilling 5 gallons of beer. (Well, not in a reasonable amount of time, anyway).
My personal opinion is that you would be better off spending that time and effort on a fountain pump and a bucket of ice water like a lot of other people. It's a proven technology and pretty inexpensive.