280 and 30-06 recoil shooting 150gr bullet is the same. If you want reduced recoil and a caliber that is effective on deer - look at 7mm-08 or 6.5x55 Swed. These would not be my first choice on Elk beyond 200 yards (especially beyond 300) but they will work with the right bullet put in the right place. Been there, done that as has a friend I hunt with.

280
 

It will do you just fine Just a little bigger than a 270 alot less recoil than 30-06. very flat shooter.Just a super caliber.

If reducing recoil is a medical necessity then I don't think you will be making that much of a difference with either the 270, 280 or the 30/06 (with all other things be alike or near same). I own all 3 & recoil depends more on the stock & pad than the recoil difference between these 3 calibers. Several years ago I also had 25/06 & it is a very easy recoiling cal, but I personally wouldnt depend on it for elk!

I have another caliber that I LOVE & I think it just might be "what your Dr. Ordered" (THANK YOU, I'll be here all night! :p ) I would SERIOUSLY recomnend you consider the 7-08! :barmy:

It's nearly TO SWEET of a caliber, yet with 150-160 monolithic type bullets, it will send ANY elk within 1/4 mile to the big wallow in the sky! :s4:
HL

http://www.chuckhawks.com/recoil_table.htm

All things being equal,stock design,recoil pad,ect... you might notice less recoil with the 280. Check out the recoil table above it may give you some other options as well.

I know this is oh so passe', but the 7x57 is a great choice too. Most rifles will handle all the bullets from 100 to 175gr . While I'm about to get flamed, it operates at lower pressures than both the 280 and the 7-08, so there's generally less muzzle blast and lower recoil. It lags a little behind the 06' but w/the modern factory offerings it becomes a real contender in what's a 100 ft lbs game.

I've a 1908 Mauser and a Savage 110 in 7x57 both are a pleasure to shoot.

I wish I could remember where I read it but a guide up in Alaska said the 7x57 was his favorite for dropping caribou and outperformed any other cartridge he'd used for the same purpose.
For my own taste I prefer the 8x57 but that definitely wouldn't serve OP's purpose for less recoil.

I cannot tell the difference in recoil in a 270 280 or 3006