Be sure the .308 is legal for bison where you hunt- won't cut the fat in Alaska.
I think.... I've been applying unsuccessfully for 40 years now, so have given up keeping current on some bison regs. Sufficient unto the day and all that (I have it caliber covered, anyhow...)..
If "the first two hurt too much to shoot" you might consider some mitigation options - premium recoil pads, muzzle brakes, double up on the ear protection (YES, it does make a difference in perceived recoil - on the bench at least!!!!) Maybe a mercury tube inletted into the fore-end.
My Ruger .338 comes close to 10 lbs weight with Mag-Na-Porting, and Decelerator recoil pad. (Said pad not even being the the best at reducing felt recoil.) My .338 has less recoil than my '06's, including muzzle jump. But those are about 2 lbs lighter, with identical recoil pads and no Mag-Na-port. Muzzle jump and noise are a huge part of perceived recoil. I seldom shoot the .338 from the bench without doubled up hearing protection. And then it is generally 1-3 shots to check/adjust zero (I know how it shoots!), and go kill something with one shot and an insurance shot, clean gun, and put away for a year or two...
I don't find this regimen painful at all....
Given what I know of the gun and the loads it likes, I'd push it to 500 yards if needed. I shoot moose at under 100 yards, mostly, and carry the .338 because I hunt moose in brown bear habitat. .338 is more comforting than an '06 when staring down at 2 gallons of steaming bear poop.......
. But the '06 is every bit as adequate on moose, as I proved a half dozen times over in the earlier years, before I got chicken.
I use other lighter/smaller caliber rifles for longer range, smaller animal prospects in more open country...