Would you guys take a steak out, and hang it in a tree for a week with cheese cloth over it, and then throw it on the grill? I know i wouldn't. But to each his own..
This is my thoughts, and it has worked well for me. I like to get the meat cold asap but not frozen. What i have done, and has produced meat that even people that don't 'eat venison' enjoy is the following...
Shoot it, gut it asap, i take a tarp lay it in my cargo hitch rack, and line the bottom with ice bags, fill the inner cavity with ice bags, and then on the outside of the deer, and one bag between the legs to get that as cool as possible, surround the neck ect ect. Then i fold the tarp over and ratchet strap it to the rack and off i go. I make sure that the bottom drains off so the carcass does not sit in water. Also take note, RETIGHTEN your ratchet straps often! ice melts quick and slacks up the straps.
When i get home I then hang it upside down, skin it, and take off the front legs whole(cutting off/throwing out below the 'knee'), those go into a cooler with ice cubes, next i cut off the back strap/tenderloins, neck roast, rear legs i cut off in one huge chunk of meat, then the rib meat ect ect. I make sure to layer with plenty of ice. DO NOT PUT IT IN PLASTIC BAGS this will trap the warmth in and act like an insulator. Any trimmings/rib meat i put in a zip lock plastic bag but make sure it isn't in a pile, that it stays a thin layer of meat in the bag and lays flat on the ice... say fill the bag only 30% of what i could hold..for fast cooling. This is OK vs the other pieces of meat since there is no bone and it lays only 1" thick or so compared to several.
At this point i'm shot, and then fully process the deer the following day. I do not let the meat sit in cold water, i make sure the drain plug is open and the cooler is tipped towards that side. I also put a ratchet strap around it so no racoons get into it (no worries of bears where i live). If you let the meat sit in cold water it starts turning white like when you put a bandaid on your finger too tight and for too long. It doesn't look good and i think would remove the blood/flavor and replace it w/ water and not taste so hot.
In your case since you'll be out for several days i would bring some bags of ice, but block ice in the cooler will last much much longer and keep the cubes from melting. Obviously always keep the coolers in the coolest/shadiest area and if you can 'prechill' the coolers with ice before adding your final load of ice do so. I'd drain them once or twice a day and do not check on their status, you are just going to let out the cold.
Bacterial likes heat and moisture, I'll take cooling my meat down to below 40F as fast as i can, vs having the deer 'dry' but hot. If you try to use ice to cool it down w/o the deer contacting it, you are most likely going to be using plastic bags which insulate. Not good. If you use dry ice not only is that expensive but i'd say you got a good chance of the meat getting frost bite/freezer burn and tasting like poop. Unless one has a walk in cooler, where you can bring the deer in and let it hang w/ the hide off to aid in rapid cooling(this is ideal IMO), i think my method or something similar is your best bet.