I can only think of 2 pronghorn that I've ever put in a cooler on ice, both in New Mexico and the temperature was in the 90's.
Never have put one on ice from Montana or Wyoming that I've shot. Never have had a bad tasting one either. I always quarter them though, and they cool very quickly.
I don't think there's anything wrong with putting them on ice in a cooler, but IMO, you have to be more careful doing that. There is zero for air circulation in a cooler, its always damp, and a prime place for bacteria growth and spoilage.
I've had more luck maintaining air flow/circulation and keeping the meat dry and letting a "skin" form on it. As long as the quarters cool out right away, the air temps at night are getting into the 40-50 range, and you keep the meat in the shade during the day...you wont have any problem.