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I have a 7400 Carbine in .30-06 and a BAR in 7mm Rem.Mag. w/BOSS-CR.
Two very different guns for different purposes for me.
The Remington is accurate enough and meant to be a versatile hunting gun. I will be switching to 150-gr. ammo to keep from pounding it to an early death. Also, a smaller scope is going on to keep it useful as the short and handy powerhouse it is.
The Browning was meant to be my nicest rifle, and it still is.
I like the 7mm Rem.Mag. and wanted the BOSS (correctly printed with periods as B.O.S.S., but that gets tiring!).
It is nicer overall than the 7400 and it's more accurate. Smoother, too.
If you got a .308 with sights, that would be a fine firearm.
If you opt for the BOSS, the new .300 WSM looks like it'll be around a while and would give you power with a capital 'P', plus the BOSS would tune it in to give you dead nuts on accuracy with your choice of ammo, not the rifle's choice. Something to think about.
If you're looking to have iron sights, then the BOSS is out.
A NIKON 2-7 scope would be a good fit and is a clear scope.
If you opt for the BOSS, any scope, large or small, can be your choice.
Mounting on a Weaver style base/rings would allow quick R&R of the scope. Weaver brand mounts/rings may not be the most aesthetic. Kwik-Site rings are good looking, though.
(check out the Beretta 1201FP shotgun for a 12-ga. autoloader that is a practical twin to the BAR in profile and operational characteristics re safety location and bolt closing button location...it's eerily similar. If I got the regular buttstock for my 1201FPG3, it would be fun that way.)
(Beretta 1201FP shotgun photo for profile comparison)