I somewhat disagree with most of the comments about the 6.5x55 being suitable for ALL the game listed. A lot depends on how good of a hunter you actually are. Can you close the distance before shooting and trail a wounded elk though the type of terrain you'll be hunting? I've never hunted elk in the NW but can assure you it might be a difficult trailing job in the dense, wet forest. Shooting an elk or moose on snow and following it up is a whole different situation.
Hitting a 9" gong under best scenario range conditions means your true hunting range should be limited to 1/2 that.
The 243 Win. is an adequate deer round but does depend on bullet construction more than a larger caliber. After some failures upon hitting a shoulder, we started using premium controlled expansion bullets in the 243 and had fewer problems in that respect. My daughter uses a 243 with Speer GrandSlam 100 grain or Nosler Solid Base 100 grain with good results. The 6.5x55 is a better deer round and you already have it.