How do you package it in the cooler and do you have issues with the meat turning grey? When a cooler is optional for me (rare) I try to use dry ice to avoid soaking in cooler juice and turning the meat grey.
This is essentially what I do with all my whitetail too, because usually it ends up too warm to hang them more than 24-48 hours. So they're in a cooler too while it takes me 7-10 days to get them processed / wrapped / frozen. So, it's worked on probably 20+ animals for me. Ideal scenario I'd have the whole animal or large pieces of meat in a walk in cooler, just not logistically possible
Let meat dry as much has possible to get the blood out. Get it to the truck - game bags go into trash bags, double bagged and knotted so they're air/water tight and are not soaking in cooler water. Trash bags go into the cooler and get completely covered in ice / ice water for the trip home.
Get home - get the meat out, drain off all blood, pat dry with paper towels, initial cleaning of dirt / whatever. Repackage meat in large ziplocs or larger trash bags, force out as much air as you can and seal them again, back in the cooler covered in ice, again - air/water tight.
For processing - I pull out a bag at a time, clean, trim, process freeze until I work through the whole animal. I have not once lost any meat due to spoilage, and the Elk I've shot aren't too far off the tenderness of a 2.5 year old whitetail (No, they weren't mature bulls....). Longest it's been is two weeks on ice using this method before freezing.
Everyone always says "Avoid moisture at all costs" - but it hasn't mattered for me as long as I get the meat to 32 degrees and keep it there, and periodically drain off the blood, and keep them from soaking in water.