I drive a Belgian .30-06 BAR, and it's a real sweetheart. But it's nowhere in the same class as a Remington 700PSS. The thinner barrel and forend attachment are intended for a sporting rifle, not a tactical, target, or varmint rifle.
I wouldn't necessarily call the Browning BAR a gun that you'd want to run a ton of milsurp .308 through, either. You'd be better off with a FAL, CETME, or VEPR, guns that are intended to be shot with military ammo, and a lot of it over their service lives.
My BAR shoots into about 1" at 100 yards with the same 168gr HPBTM handloads I make for my 1903's and Garands. I've had to use small-base dies for these handloads, because the Browning has a very tight chamber nearest the bolt. It's not a bad thing, but you have to be careful. Getting sub MOA groups from milsurp ammo may not be that easy from a BAR. The fact that the Remington 700 PSS can do it is a tribute to the gun's inherent accuracy.
Granted, the BAR feels good and shoots nicely for an autoloader, but it wouldn't be my first choice for a milsurp blaster. They're an awfully nice gun to be reduced to that purpose. Unless, of course, money isn't an object after all...