Rod, I would not hesitate to tote any of those (6.5X55, 7X57, .257 Roberts) and be perfectly confident in the shot.
As for the Nosler Accubond, I took SSA's 110 gr offering at 2650fps, using a BC of .370 from their website, and plugged it into handloads.com's ballistic calculator. At 350, you're at 857 and only 70 fps above the minimum recommended limit for reliable expansion.
At 150 yards, I'd saddle up right next to you with my 6.8!
As far as the .257, Remington's 117 gr loading is holding 999 ft lbs at 200 yards, and drops to ~600 ft pounds at 350 yards. I'd stop at 200 with it.
For the 6.5X55, at 400 yards, you are still pushing 1,023 ft lbs and 1814 fps with the Remington 140 gr offering! That's why the Europeans love that round!
As for the 7X57, you are still cooking with 1,037 ft pounds and 1827 fps with the Remington 140 gr load.
So, comparing them to the 6.8 just doesn't make sense.
But, again, I assert that if you are really concerned with rapid follow up shots with an AR platform, and want the 350 yard capability, look into the AR-10 family (using the AR-10 to denote all .308 sized platforms). I've got the DPMS LR-308B and shooting quick follow ups is NOT a problem. You'll be out close to a grand for that model, not much farther off than if you were to buy a Grendel or 6.8 (not considering kits).