I'm in the middle of a suppressed 7mm-08 build for precisely this reason. For what it's worth, I'd grab a Tikka T3x Lite, get the barrel shortened to 18" and threaded, add a suppressor, and dial it in with Barnes 120-grain TTSX. Plenty of power for elk within reasonable range and the recoil of a .243. Lots of good suggestions here, you'll find what works for you. Good luck!
