Low recoil rifle for elk, deer, etc. but .270 or larger caliber

Not really, no. A 150 grain out of a 7-08 will a) run about 2770 vs 2875 for the 308, and b) burn a fair bit less powder to do it. Using reloading data, we get:

7.5 lb rifle in both cases, and using kwk.us/recoil.html to calculate:

150 grain 7mm-08 at 2770 FPS using 40 grains of powder = 14.3 lbs of recoil (12.7 in an 8.5lb rifle).

150 grain 308 at 2875 FPS using 47 grains of powder = 17 lbs of recoil (15 in an 8.5 lb rifle)

So the 7mm-08 is around 16% less recoil. That's plenty less recoil to be noticably less, both on paper and in real life (I've owned Tikka's in both 308 and 7mm-08, and the 7mm-08 was very noticeably less recoil). There will be some variation in those recoil numbers depending on what powder factory ammo used, those are just what I use for my reloads (Varget in both cases), but not all that much.
I’m always open to learning. I see from examining Hodgdon tables for .308 and 7-08 that you are about 1/3 of the way up between min and max loads for varget on the 7-08, but maxed out on the .308 load, so of course more recoil would be expected using a 150 gr projectile. For 7-08, I use Staball 6.5 at 46gr behind a 150 ELDX which delivers more pressure and correspondingly puts the projectile in the same velocity class as the .308 with varget. What I should have said is “if both projectiles are launched at the same velocity” then recoil should be almost identical. I should note that I don’t use the max listed load of 48.5 gr because I get pressure signs. (This is out of a tikka T3x)

Am I correct in assuming that the .284 projectile with a higher BC will have flatter ballistics and maintain more velocity at 500 than would a .308 projectile at the same velocity?
 
I’m always open to learning. I see from examining Hodgdon tables for .308 and 7-08 that you are about 1/3 of the way up between min and max loads for varget on the 7-08, but maxed out on the .308 load, so of course more recoil would be expected using a 150 gr projectile. For 7-08, I use Staball 6.5 at 46gr behind a 150 ELDX which delivers more pressure and correspondingly puts the projectile in the same velocity class as the .308 with varget. What I should have said is “if both projectiles are launched at the same velocity” then recoil should be almost identical.

Am I correct in assuming that the .284 projectile with a higher BC will have flatter ballistics and maintain more velocity at 500 than would a .308 projectile at the same velocity?
Im also open to learning, but as a guy who reloads for both 7mm08 and .308, I havent seen a combination yet where a
308 maintains more velocity than the 7mm08.

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The likelihood of you drawing a tag in KY is slim to none. You have roughly less than 1%...... So I would personally just buy whatever you want, then IF you draw a tag in KY, get a rifle that is .277 cal or bigger.
 
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