its a pain to get the brains out while in the field. You will need a coat hanger, or something similar to scramble the brains and have them come out the small opening, and a way to rinse it if possible. without water pressure from a hose or something similar, getting them all out will be difficult.
So yes, keeping it cool to keep it from getting smelly while out for a week or more should help.
Same as everyone else. I do my own skulls. Very easy if you have a pressure washer. Heat in water with dawn dish soap and oxy clean, remove 90% of flesh with pressure washer, 30-60 mins of scraping, dump in bucket with bleach and peroxide. Easy day project, almost zero cost.
Remove as much meat as possible along with the eyes and all the fatty tissue you can from behind them. Also make sure to get all the skin and hair of around the bases of the antlers just so it doesn't harden like a rock. That will be rehydrated if boiling, but can be pesky if your guy uses beetles. If you have a piece of wire (from a coat hanger) you can bring with to scramble the brains that would be helpful because you can make a small hook on the end to grab pieces of the brain that stay stuck inside because of a membrane tissue. A little time with that and small aggressive back and forth movement in some water source should be able to remove everything needed. I agree with the other post that if your successful you'll need something to do at camp and in no time it should be pretty clean once you leave the field.
Cape it off the skull. I'm the kind of guy as to where, if I have the time, I practice caping out my skulls. After the cape is off remove the lower jaw bone, any excess meat, eyes, and as much brain matter as possible. Then I usually wrap a garbage bag over the skull and tape around the bases of the horns. Keeps the bugs from getting into it.