I have been a 6.5 shooter for close to two decades using the .260, which has a small velocity edge on the 6.5CM. I would not recommend either for Elk beyond 300-400yd.
The critical issue is not about shot placement; and choosing a specific chambering because of accuracy is missing the point.
The critical issue with hunting bigger game is terminal energy delivery. A hunter should be able to deliver the shot ontarget regardless of chambering, and only then does chambering choice make the difference, specifically in delivering adequate energy to that point of impact.
The same rifle is available in .30-'06, and I would much prefer that chambering out of respect for the animal; it deserves a swift and reliable kill. If you're not handloading, then at least invest the time and effort into the accuracy evaluation of available factory ammo that is built around a bullet with reliable expansion at the expected impact velocities.
I'd start my testing with
Hornady Precision Hunter 30-06. It's a cruise missile with serious energy delivery capability.
I know from my own experience that the .30-'06 can deliver significantly sub-MOA accuracy performance at 200yd out of a 22" sporter barrel in a Winchester Model 70 Featherweight. The handload used is Rem brass, 48.1gr of IMR-4064, Win WLR Large rifle Primer, and the Sierra 165gr GameKing. Your results might vary, but I don't think by much; as the same load with the 168SMK was developed as a clone for Federal FGMM 168gr Match, and performs as intended. That rifle and load (168FGMM) took 4th place in the NJ State Sniper Championships in 1997 shooting off a bipod. No, it's not a max load; it's an adequate load with impressive accuracy, which is what I think you're actually looking for. On deer, it flat out works for me.
The topic's subject rifle looks like a very good one; and for Elk, the chambering escalation should be a reasonable choice. Recoil is going to be a factor, but should not be an issue with a hunting rifle. I suggest .30-'06 because I believe it is a good compromise between terminal performance and recoil. The real issue with recoil is that too much discourages the volume of practice that a good shooter needs to engage.
My own choice of deer rifle is a Model 70 .30-'06 Featherweight, and for Elk, the chambering would not be a smaller one.
Greg