We had our goats on ice for three days before we processed them at home. I would feel comfortable going a week at least as long as I was diligent about topping off the ice and draining the water. If your cooler is good, after a day or two the ice hardly melts. Cheaper coolers require more maintenance.
I also agree about the taste of goats, given the preference, shoot one on irrigated crops. Not a huge deal, but if you or your family is picky and don't like the true taste of wild game then young animals on crops are the way to go. Young antelope on grain are pretty much the most mild meat out there, less flavor than beef.