We have been doing the 1000 to 1500 mile trips from Western Oregon to the mountain states for around 30 years.
We almost always have the meat processed locally (near where we kill the animal) while we spend a day or two recuperating and exploring the area. Many butchers in Wyoming will cut the next day after it has chilled overnight in their cooler (it's a pain to cut meat that hasn't been chilled and firmed up). After the meat is cut and wrapped, it needs another overnight to be frozen, then it's ready to pick up so roughly a 36-48 hour turn around. I like to call meat processors before I leave for the hunt and see if they can do a fast turn around since I got burned once on this in New Mexico.
If we can't get the meat to the processor immediately and it's too warm outside, we will put the meat in large garbage bags and submerge it in ice. This is fairly easy to do with a deer or an antelope, but a bit of a challenge with an elk due to the sheer size.
We bring the processed meat back home frozen with dry ice to keep it frozen. Walmarts tend to have dry ice. Use a towel or some wadded up brown paper bags to buffer the meat from the dry ice so it does not freezer burn. Ice chests work great as does a small chest freezer. We have a small chest freezer that fits on the tongue of our ATV trailer.
The last two elk we have had processed cost $165 for a nice sized cow (locally here in Oregon) and around $200 for a 6 point bull in Wyoming. Both of those prices I felt were on the low side. I have seen prices for an elk up to around $450. Some places charge by the pound of hanging weight, some by the animal.
In Cody I have been given the names Wagler Meats and Zero Box Meats as places that process game.
This is our cutting instructions for an elk:
Make EVERYTHING possible into steaks, 3/8" thick. My kids LOVE an elk steak dredged in seasoned flour and fried hot in butter to medium rare. From the drippings we make a gravy and serve it all with pancakes (kind of like biscuits and gravy, but pancakes instead). That is our weekend family favorite breakfast. One pound packages.
Keep tenderloins whole, package separately.
Backstraps: Keep in whole chunks (about 3 to 4 chunks per elk backstrap and 2 for deer or antelope). This give you the most options of how to prepare this, including cutting into steaks later if you like.
Burger: No fat added, one pound packages
A few processors vacuum pack meat in plastic bags and this is the very best. The meat keeps a long time and it doesn't drip in the refrigerator when it is thawed. Most butchers however use butcher paper.