Ahh, you didn't mention that. I'd go with the .270 then. The .26/06 would probably be "okay" for Elk but not really one I would choose over the .270. The main problem I am seeing is your wanting to cut the barrel to 22" which will hinder the performance of any factory ammo for either choice you have listed. Handloading would be a different story but still not that much different. That's a good bit of case capacity for a medium to quarter bore and just 22" of length to burn it. Were it me, I would see about trading that action for a short action and go with either a .260 , or a 7mm.08 for that short barrel. Or, you can go really sporty and go with a .338fed for the big bore rockum sockum hello Dolly stuff Or you can do as you suggested, switch bolt faces to a 0.534 and go with 6.5 Remington Mag, 7mm WSM, 270 WSM, 300 WSM, 350 Remington Mag, 338-300 WSM, 450 Marlin, 325 WSM. The list is fairly long but that is shortened by your wanting a shorter barreled rifle.

Edit to add: I just went through this same thought process myself in making the decision for an AR build. NoMercy had me switch my brain on about short barrel performance. It's the main reason I went with .338fed because of the performance levels I could reach with a shorter barrel length (went with 22" Black Hole barrel)