I've put down a couple younger elk, 2-3 years old, and have found a yeti 105 and 65 was enough room for all the boned out meat of one elk. I think a big help of getting the meat cooled down is having your cooler already cold. I travel from Michigan to MT and ID for elk, before I leave, I have both coolers full of frozen 1 gallon jugs of water. I'll add ice to fill in the gaps. Keep draining the melted ice out of your cooler, it's good for rinsing hands, face, dishes at the truck if you've been day hunting. The nice thing about the frozen water jugs is as they are melting, you have safe ice cold drinking water that doesn't need filtering. Unless its abnormally warm, those two coolers will stay cold and have a mix of water and ice in the water jugs after 7-8 days. When getting my meat into the cooler, I drink/drain the water jugs until they are just ice and then squeeze the air out of the jug, cap it, and placed the compressed jug with ice chunk inside it in between quarters. Great way to cool meat down.
Wish I could take credit for this process but this is all from a Randy Newberg podcast.