I don't see any problems with submerging the whole animal to give it an intial quick cooling but once you cut into the skin exposing meat, do not get water onto it! The trick to preserving meat is to keep it"Cool and dry". You can use plastic bags to dunk pieces into the stream but really, how cold is the water? It may be cold enough to cool it down a little but the juices will drain from the meat into the bag and then you have wet meat that is cool but not cold, therefore another bad situation arises. I would keep a cooler of ice in your truck, gut theelk quickly tofacilitate internal coolingand get it to the ice as soon as possible. Definitly bag it in cloth meat bags and hangthem in a shady place for the meantime. Might be a good idea to salt the meat in order to buy more time (common practice here) and pepper everything to keep flies from contaminating it.

I have another trick to throw out. Find a crack in the rocks and find some kind of plant whose leeves are clean and feel cool to the touch, you know what I mean? Fill the crack with the leaves, place your meat or meat bags on them and cover with more leaves. I was searching interviews with Hawaiian shore fishermen from the early 1900s and found that they used this method with ti-leaves to preserve fish in the heat. They called it a "cool box."