My hunting is very similar except I throw in a moose every now and then. For this range, I wouldn't bother with a RUM. Too much of a good thing and I don't like carrying a 30" barreled rifle in the woods.
The difference is slight in 24" barrels when compared to the WSM/Win Mag.
My personal walk about rifle is a Savage 110 270. Used to be a '06 but like the better ballistics. Going to 600yds is no issue and the rifle is accurate enough to make the journey...once. I use 150gr SST.
My cut block/stand rifle is a 6.5-06. Launching 140gr SST, again no issue within these ranges. Plus, the BC is much better then the 150gr 270. Hitting steel gongs at 750yds shows me these 140gr SST will have no issue laying any game I want in the back of my truck. I have a muzzle brake
on this so recoil is pretty much zip. I need to spot my own shots. Rifle is 12lbs in a thumbhole lam stock. Not so portable.
Because I am a continuous tinkerer, I now have a Savage in 7RM. Muzzle brake installed and I hope to fireform some brass this week. This one will be loaded with 162gr SST/Amax at around 3100fps, I hope. This is just more of a good thing and will allow me to hunt in a few locations where shots will go to 900yds. Rifle should weigh around 9lbs so easily portable to those secluded cut blocks. not so sure about the recoil side. Time will tell.
Is the 7RM any better then the 270 or 6.5? NO but I like toys. Ballistically, all pretty close with the 7mm having the most punch. Will any animal care? Will your shoulder and ears?